Ramseytype theorems with forbidden subgraphs


 Tiffany Reynolds
 2 years ago
 Views:
Transcription
1 Ramseytype theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called Hfree if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal. Is it true that for every graph H, there exists a ε(h) > 0 such that ay Hfree graph with vertices cotais either a complete or a empty subgraph of size at least ε(h)? We aswer this questio i the affirmative for a special class of graphs, ad give a equivalet reformulatio for touramets. I order to prove the equivalece, we establish several Ramsey type results for touramets. 1 Itroductio Give a graph G with vertex set V (G) ad edge set E(G), let α(g) ad ω(g) deote the size of the largest idepedet set (empty subgraph) ad the size of the largest clique (complete subgraph) i G, respectively. A subset U V (G) is called homogeeous, if it is either a idepedet set or a clique. Deote by hom(g) the size of the largest homogeeous set i G, i.e., let hom(g) = max (α(g), ω(g)). If H is ot a iduced subgraph of G, the we say that G is a Hfree graph. Accordig to classical Ramsey theory, hom(g) 1 2 log 2 for every graph G with vertices [ES35], ad there exists some G with hom(g) < 2 log 2 (see [E47]). Erdős ad Hajal [EH89] raised the possibility that the followig could be true. Departmet of Mathematics, Raymod ad Beverly Sackler Faculty of Exact Scieces, TelAviv Uiversity, Tel Aviv, Israel. Supported by a USA Israeli BSF grat, by a grat from the Israel Sciece Foudatio ad by the Herma Mikowski Mierva Ceter for Geometry at Tel Aviv Uiversity. Mathematical Istitute of the Hugaria Academy of Scieces H1364 Budapest, P.O.B Supported by NSF grat CR , PSCCUNY Research Award , ad OTKAT Computer ad Automatio Istitute of the Hugaria Academy of Scieces H1518 Budapest, P.O.B. 63. Supported by TKI grat ad OTKAT
2 Cojecture 1 For every graph H, there exists a positive ε = ε(h) such that every Hfree graph with vertices has a homogeeous set whose size is at least ε. Erdős ad Hajal cofirmed their cojecture for every graph H which belogs to the class H defied recursively as follows: 1. K 1, the graph cosistig of a sigle vertex, belogs to H; 2. if H 1 ad H 2 are two vertexdisjoit graphs belogig to H, the their disjoit uio as well as the graph obtaied from this uio by coectig every vertex of H 1 to every vertex of H 2 belogs to H. Gyárfás [G97] oticed that it follows from a well kow result of Seische [Se74] that Cojecture 1 is also true for all graphs geerated by the above rules startig with P 4, a simple path with 4 vertices, ad K 1. Our first theorem exteds both of these results. If Cojecture 1 is true for some graph H, the we say that H has the ErdősHajal property. For ay graph H with vertex set V (H) = {v 1,..., v k } ad for ay other graphs, F 1,..., F k, let H(F 1,..., F k ) deote the graph obtaied from H by replacig each v i with a copy of F i, ad joiig a vertex of the copy of F i to a vertex of a copy of F j, j i, if ad oly if v i v j E(H). The copies of F i, i = 1,..., k, are supposed to be vertex disjoit. Theorem 1.1 If H, F 1,..., F k have the ErdősHajal property, the so does H(F 1,..., F k ). I other words, the ErdősHajal property is preserved by replacemet. This eables us to verify that Cojecture 1 is true, e.g., for the graphs depicted i Figure 1, which aswers some questios of Gyárfás [G97]. Figure 1: G 1 ad G 2 have the ErdősHajal property No operfect graph is kow to have the ErdősHajal property. Ufortuately, i this respect Theorem 1.1 caot offer ay help. Ideed, accordig to a result of Lovász [L83], which played a key role i his proof of the Weak Perfect Graph Cojecture [L72], perfectess is also preserved by replacemet. It is a outstadig ope problem to decide whether the smallest operfect graph, the cycle of legth 5, has the ErdősHajal property. As Lovász poited out, there is a eve simpler usolved Problem Does there exist a positive costat ε so that, for every graph G o vertices such that either G or its complemet Ḡ cotais a iduced odd cycle whose legth is at least 5, we have hom(g) ε? 2
3 It is easy to formulate aalogous questios for touramets. A touramet with o directed cycle is called trasitive. If a touramet has o subtouramet isomorphic to T, the it is called T free. It is well kow [EEH73],[S74] that every touramet of vertices cotais a trasitive subtouramet whose size is at least c log, ad that this result is tight apart from the value of the costat. Cojecture 2 For every touramet T, there exists a positive ε = ε(t ) such that every T free touramet with vertices has a trasitive subtouramet whose size is at least ε. Theorem 1.2 Cojecture 1 ad Cojecture 2 are equivalet. I order to prove Theorem 1.2, we eed a Ramseytheoretic result for touramets, which is iterestig o its ow right. A touramet T with a liear order < o its vertex set is called a ordered touramet ad is deoted by (T, <). A ordered touramet (T, <) is said to be a subtouramet of aother ordered touramet, (T, < ), if there is a fuctio f : V (T ) V (T ) satisfyig the coditios (i) f(u) < f(v) if ad oly if u < v, (ii) f(u)f(v) E(T ) if ad oly if uv E(T ). Theorem 1.3 For ay ordered touramet (T, <), there exists a touramet T such that, for every orderig < of T, (T, <) is a subtouramet ( of (T ), < ). Moreover, if T has vertices, there exists a T with the required property with O 3 log 2 vertices. We further show that the O( 3 log 2 ) estimate is ot very far from beig tight. I fact, if (T, <) is ay touramet o vertices ad T satisfies the coditio above for T, the T must have at least Ω( 2 ) vertices. The proof of the above theorem is very similar to the proof of the mai result of [RW89], which deals with a similar statemet for ordered iduced subgraphs. This ca be exteded to hypergraphs as well. By choosig a bigger touramet T, oe ca esure a sigle touramet that cotais all ordered touramets o vertices, i ay orderig. Specifically, we prove the followig. Theorem 1.4 Give a iteger N, let 0 be the largest iteger such that ( ) N 2 ( 0 2 ) 1, 0 ad put = 0 2. The, for all sufficietly large N, there exists a ordered touramet T o N vertices such that i ay orderig it cotais every ordered touramet o vertices. 3
4 Note that the above estimate for is clearly tight, up to a additive error of 2. A similar statemet holds for iduced subgraphs, as show i [BK93]. The rest of this paper is orgaized as follows. Theorem 1.1 is proved i Sectio 2. The proofs of Theorems 1.3 ad 1.4 appear i Sectio 3. Sectio 4 cotais the proof of Theorem Graphs with the ErdősHajal property I this sectio we prove Theorem 1.1. Obviously, it is sufficiet to show the followig weaker versio of the theorem. Theorem 2.1 Let H ad F be graphs havig the ErdősHajal property, V (H) = {v 1, v 2,..., v k }. The the graph H(F, v 2,..., v k ), obtaied by replacig v 1 with F, also has this property. Proof: Let H 0 deote the graph obtaied from H by the deletio of v 1. For simplicity, write H(F ) for H(F, v 2,..., v k ). Let G be a H(F )free graph with vertices, ad assume that hom(g) < ε(h)δ. We would like to get a cotradictio, provided that δ > 0 is sufficietly small. Let m := δ > k. By the defiitio of ε(h), ay melemet subset of U V (G) must iduce at least oe subgraph isomorphic to H. Otherwise, we would fid a homogeeous subset of m ε(h) > hom(g) i the subgraph of G iduced by U, which is impossible. Therefore, G has at least ( ) ( m / k m k) iduced subgraphs isomorphic to H. For each of these subgraphs, fix a isomorphic embeddig of H ito G. Sice the umber of embeddigs of H 0 ito G is smaller tha ( 1) ( k + 2), there exists a embeddig, which ca be exteded to a embeddig of H i at least ( m) M := ( k ) (1) m k ( 1) ( k + 2) differet ways. I other words, there are k 1 vertices, v 2,..., v k V (G), ad there exists a at least Melemet subset W V (G) such that, for every w W, f(v 1 ) = w, f(v i ) = v i (i = 2,..., k) is a isomorphic embeddig of H ito G. Cosider ow the subgraph G W of G iduced by W. This graph must be F free, otherwise G would ot be H(F )free. Sice F has the ErdősHajal property, we kow that hom(g W ) W ε(f ) M ε(f ). O the other had, ε(h)δ > hom(g) hom(g W ). 4
5 Comparig the last two iequalities ad pluggig i the value (1) for M, we obtai that ( δε(h)/ε(f ) > ( m) k + 1 k ) = m k ( 1) ( k + 2) m(m 1) (m k + 1) > 1 kδ, which gives the desired cotradictio, provided that δ < ε(f ) ε(h) + kε(f ). 3 Ramseytype theorems for touramets The proof of Theorem 1.3 uses the probabilistic method. The basic idea is a slightly simplified versio of the mai argumet of Rödl ad Wikler i [RW89]. We eed the followig lemma. Lemma 3.1 Let t > > 1 be two positive itegers, ad let S = {a 1, a 2,..., a t } be a telemet set. Let g : S R = {1, 2,..., t} be a fuctio such that for every p R, we have {i : g(a i ) = p} =. Further, let f : S N = {1, 2,..., } be a radom fuctio obtaied by choosig, for each elemet a i S, radomly, idepedetly, ad with uiform distributio a value f(a i ) N. Let E be deote the evet that there exist 1 i 1 < i 2 <... < i t such that g(a ij ) g(a ik ) for all 1 j < k, ad f(a ij ) = j for all 1 j. The the probability that E does ot hold is at most 1 q=0 ( ) t q( 1) ( 1) t q q t q=0 ( ) 4et e t. Proof : To estimate the umber of fuctios f for which the evet E fails, we argue as follows. Give such a f, let i 1 be the smallest iteger (if it exists) such that f(a i1 ) = 1. Assumig i 1 < i 2 <... < i j 1 have already bee defied, ad assumig that f(i s ) = s for all s < j ad that the elemets g(a is ), s < j, are pairwise distict, let i j be the smallest iteger (if it exists) satisfyig i j > i j 1, f(a ij ) = j ad g(a ij ) g(a is ) for all s < j. Note that, sice the evet E fails, this process must termiate after some q 1 elemets i s have bee defied. Note also that if k is a idex satisfyig i s 1 < k < i s, ad g(a k ) differs from g(a ij ) for all j q (or eve just for all j < s), the f(a k ) caot be equal to s (sice otherwise we would have defied i s = k). Sice there is a similar restrictio for the value of f(a k ) for k < i 1 ad for k > i q, it follows that oce the sequece i 1 < i 2... < i q has bee defied, the value of f(a k ) ca attai at most 1 values for all but at most t q elemets a k. Therefore, the total umber of fuctios f for which the evet E fails is at most ( ) 1 t q( 1) ( 1) t q. q 5
6 Sice the total umber of possible fuctios f is t, the probability that E does ot hold is at most ( ) 1 t q( 1) ( 1) t q 1 ( ) et q ( ) q ( 1 q t q ( 1) q 1 1 ) t q=0 q=0 ( et (1 + 1 ) 1 ) e t ( ) 4et e t. Proof of Theorem 1.3: Let (T, <) be a ordered touramet o the set N = {1, 2,..., } of vertices, ordered aturally. We may ad will assume that is sufficietly large. Let c > 3 be a absolute costat, ad let t be the smallest iteger satisfyig t > c log such that t 1 is a prime. By the kow estimates for the distributio of primes, t = (1 + o(1))c log. Let P be a projective plae of order t 1. Each lie of P cotais precisely t poits, ad the umber of poits i P is (t 1) 2 + t < t 2. Replace each poit p P by a set S p of poits, where all sets S p are pairwise disjoit. Costruct a touramet T o the set p P S p of less tha t 2 vertices as follows. For every lie l i P, let f l : p l S p N = {1, 2,..., } be a radom fuctio, where each image f l (u) is chose radomly, uiformly ad idepedetly i N, ad the fuctios correspodig to differet lies are chose idepedetly. For u, v p l S p, where u S p, v S p ad p p, let uv be a directed edge if ad oly if f l (u)f l (v) is a directed edge of T. The edges with two edpoits i the same set S p are orieted arbitrarily. To complete the proof, we show that almost surely (that is, with probability tedig to 1 as teds to ifiity), T cotais a ordered copy of T i ay orderig. Fix a orderig < of T, ad let us estimate the probability that i this orderig (T, < ) cotais o ordered copy of T. For each lie l i the projective plae, the orderig < iduces a orderig of the t vertices p l S p. Let S = (a 1, a 2,..., a t ) be this iduced orderig. Defie g(a i ) = p if a i S p. The, for every p l, {i : g(a i ) = p} = S p =. Observe ow that, by Lemma 3.1, the probability that (T, <) is ot a subtouramet of the ordered subgraph of (T, < ) cosistig of all edges ruig betwee distict groups S p (p l), is at most ( 4et ) e t. This follows from the fact that, if the evet E i Lemma 3.1 holds for f = f l, the a i1,..., a i iduce a copy of T, as required. Sice the evets for distict lies are totally idepedet, the probability that (T, < ) cotais o ordered copy of (T, <) is at most ( ( 4et ) (t 1) 2 ) e t +t = e (1+o(1))c3 3 log 3. The total umber of orderigs of T is (((t 1) 2 + t))! e (1+o(1))3c2 3 log 3, ad as c > 3, by our choice, the probability that T fails to cotai a copy of T i some orderig is o(1), completig the proof. We ext show that the O( 3 log 2 ) upper boud caot be replaced by o( 2 ). We eed the followig wellkow result. 6
7 Lemma 3.2 ([D67], [A68]) The umber Aut(T ) of automorphisms of ay touramet T o vertices does ot exceed 3 ( 1)/2. Theorem 3.3 There exists a absolute costat b 1 3e 2 with the followig property. Let (T, <) be a ordered touramet o vertices, ad suppose T is aother touramet such that for every orderig < of T, (T, <) is a iduced subtouramet of T. The T has at least b 2 vertices. Proof: Let N be the umber of vertices of T. The the total umber of iduced labelled (but ot ecessarily ordered) copies of T i T is at most ( N ) Aut(T ), which, by Lemma 3.2, does ot exceed ( en ) 3 /2. It follows that the probability that for a radom orderig < of T, at least oe of these copies is ordered, is at most ( en ) 3 /2 1! ( 3e 2 N 2 ). If N < 2 /( 3e 2 ), this umber is less tha 1, implyig that there is a orderig < with o ordered copy of (T, <). Thus, we have N 2 /( 3e 2 ), completig the proof. The discussio for touramets ca be easily adapted to iduced subgraphs of graphs. A simple udirected graph H with a liear order < o its vertex set is called a ordered graph ad is deoted by (H, <). A ordered graph (H, <) is said to be a iduced subgraph of aother oe, (H, < ), if there is a fuctio f : V (H) V (H ) such that, for ay u, v V (H), (i) f(u) < f(v) if ad oly if u < v, (ii) f(u)f(v) E(H ) if ad oly if uv E(H). The proof of Theorem 1.3 ca be easily modified to deal with ordered graphs, givig the followig result of Rödl ad Wikler. Theorem 3.4 ([RW89]) For ay ordered graph (H, <), there exists a graph H such that, for every orderig < of H, (H, <) is a iduced subgraph ( of (H ), < ). Moreover, if H has vertices, there exists a H with the required property with O 3 log 2 vertices. Note that there is o otrivial aalogue of Theorem 3.3, sice the umber of automorphisms of a udirected graph o vertices ca be as large as!. I fact, if (H, <) is a ordered complete graph o vertices, the the graph H = H has oly vertices ad cotais a iduced ordered copy of (H, <) i ay orderig. Combiig the above argumets with some kow results about packigs, we ca exted the last result to iduced hypergraphs as well. Moreover, the estimate for hypergraphs with o edge of size less tha 3 is slightly better tha the correspodig result for graphs. A hypergraph H with a liear order < o its vertex set is called a ordered hypergraph ad is deoted by (H, <). A ordered hypergraph (H, <) is said to be a iduced subhypergraph of 7
8 aother oe, (H, < ), if there is a fuctio f : V (H) V (H ) such that, for ay u, v V (H), f(u) < f(v) if ad oly if u < v, ad a set of vertices forms a edge iff its image uder f forms a edge. Theorem 3.5 For ay ordered hypergraph (H, <) i which each edge cotais at least 3 vertices, there exists a hypergraph H such that, for every orderig < of H, (H, <) is a iduced subhypergraph of (H, < ). Moreover, if H has vertices, there exists a H with the required property with O ( 3) vertices. Proof: Let (H, <) be a ordered hypergraph o the set N = {1, 2,..., } of vertices, ordered aturally, where each edge of H is of size at least 3. Let c be a absolute costat such that 4ece c < 1/2 (c = 5, for example, will do). Let t be the smallest prime satisfyig t > c (the t = (1 + o(1))c.) As described i [K95], there is a simple, explicit costructio of a family L of t 3 subsets of a set P of size t 2 such that each member of l is of cardiality t ad the itersectio of o two members of L is of size more tha 2. Replace each elemet p P by a set S p of poits, where all sets S p are pairwise disjoit. Costruct a hypergraph H o the set p P S p of t 2 vertices as follows. For every l L, let f l : p l S p N = {1, 2,..., } be a radom fuctio, where each image f l (u) is chose, radomly, uiformly ad idepedetly i N, ad the fuctios correspodig to differet members l L are chose idepedetly. If u 1,..., u r are vertices i p l S p, the {u 1, u 2,..., u r } is a edge of H iff the vertices u i belog to pairwise distict sets S p, ad {f l (u 1 ), f l (u 2 ),..., f l (u r )} is a edge of H. Note that, sice the itersectio of ay two distict members of L is of size at most 2, ad H has o edges with fewer tha 3 vertices, oe of the edges defied above ca lie i the uio p l S p, for ay l L, l l. To complete the proof, we show that almost surely H cotais a ordered iduced copy of H i ay orderig. Fix a orderig < of H, ad let us estimate the probability that i this orderig (H, < ) cotais o ordered iduced copy of H. For each l L, the orderig < iduces a orderig of the t vertices p l S p. Let S = (a 1, a 2,..., a t ) be this iduced orderig. Defie g(a i ) = p if a i S p. The, for every p l, {i : g(a i ) = p} = S p =. Observe ow that, by Lemma 3.1, the probability that (H, <) is ot a iduced subhypergraph of the iduced ordered subhypergraph of (H, < ) o S, is at most ( 4et ) e t 2. This is true, because if the evet E i Lemma 3.1 holds for f = f l, the the vertices a i1,..., a i iduce a copy of T, as required. Sice the evets for distict sets l L are totally idepedet, the probability that (H, < ) cotais o ordered copy of (H, <) is at most (2 ) t3 = 2 (1+o(1))c3 4. The total umber of orderigs of H is (t 2 )! e (1+o(1))3c2 3 log, ad thus the probability that H fails to cotai a copy of H i some orderig is o(1), completig the proof. It is worth otig that the argumet i the proof of Theorem 3.3 also works for hypergraphs whose group of automorphisms is ot too large. I particular, if the hypergraph H i the statemet 8
9 of the last theorem has o otrivial automorphisms, the the umber of vertices of ay hypergraph H satisfyig the assertio of the theorem must be at least Ω( 2 ). Returig to touramets, we ow describe a proof of Theorem 1.4, usig Talagrad s Iequality [T95]. A alterative proof ca be give usig the methods of [BK93]. Proof of Theorem 1.4: Let N, 0 ad be as i the statemet of the theorem, ad let T be a radom touramet o the vertices 1, 2,..., N, obtaied by choosig, for each pair of vertices i, j of T, radomly, uiformly, ad idepedetly, either the edge ij or the edge ji. Wheever it is eeded, we assume that N is sufficietly large. To complete the proof, we show that almost surely i every orderig, T cotais a ordered copy of every touramet o vertices. To this ed, fix a orderig < of T, ad fix a ordered touramet T o vertices. We use Talagrad s Iequality (see, e.g., [AS00], Chapter 7) to estimate the probability that i this orderig T cotais o ordered copy of T. The computatio here is very similar to the oe estimatig the probability that the clique umber of the radom graph G(, 1/2) is less tha its expected value by at least 2. For each set K of vertices of T, let B K be the evet that the iduced subgraph of (T, < ) o K is a ordered copy of (T, <). The the probability Pr(B K ) of each evet B K is precisely 2 ( 2). Defie µ = ( N ) 2 ( 2), ad ote that this is the expected umber of ordered copies of (T, <) i (T, < ). A simple computatio shows that the umber 0 defied i the statemet of the theorem satisfies 0 = (1 + o(1))2 log 2 N, implyig that for the fuctio f(m) = ( N m) 2 ( m 2 ) ad for every m close to 0, we have f(m + 1)/f(m) = N 1+o(1). Therefore, µ N 2 o(1). For two subsets K ad K, each cotaiig vertices of T, let K K stad for the fact that 2 K K 1. Defie, further, = K K Pr(B K B K ), where the sum rages over all ordered pairs (K, K ) with K K. Therefore, 1 2 is the expected umber of pairs of ordered copies of T that share a edge. The techical part of the proof is a careful estimate of the quatity /µ 2. Observe that if K K = i ( 2) the Pr(B K B K ) 2 2( 2)+( i 2). I fact, Pr(B K B K ) is equal either to zero, or to the righthad side of the above expressio. Thus, it follows that 1 i=2 i, where i = ( N )( i )( N i ) 2 2( 2)+( i 2). Therefore, ( N )( N ) i µ 2 = i)( i 2 2( 2)+( 2) i ) 22 2( 2) ( N 9
10 It follows that ( N ) = i)( i 2 ( 2) i ( N ) ( ( ) ( ) 2 i N )i 2 (i 2) i N 2(i 1)/2. ad that for each i satisfyig, say, 3 i 100, we have Furthermore, for 100 < i 1.9 log 2 N, we have 2 µ N 2, (2) i 6 = O( ). (3) µ 2 N 3 i µ 2 ( 2 N 0.05 )i < 1 N 5. (4) For every i i the rage 1.9 log 2 N i 1, put i = j, ad ote that 1 j (0.1+o(1)) log 2 N ad ( N )( N ) i µ 2 1 i N 2 o(1) µ = 1 j)( j 2 ( 2) j( j) ( 2) j N 2 o(1) ( N ) 2 ( 2) ( )( ) 1 N N 2 o(1) 2 j( j) 1 ( j j N 2 o(1) N2 ( j)) j ( ) 1 j 1 N 2 o(1) N 0.9 o(1) N 2.9 o(1). Combiig the last iequality with iequalities (2), (3) ad (4), we coclude that (2+o(1)) 4. µ 2 N 2 Let X = h(t ) deote the maximum umber of pairwise edgedisjoit ordered copies of T i (T, < ). We claim that the expected value of X = h(t ) satisfies E(X) ( o(1))n2 4. (5) To see this, defie p = N 2, ad ote that, by a simple computatio, we have p < 1. Let S be 2 4 µ a radom collectio of ordered copies of T i T obtaied by choosig each ordered copy of T i T to be a member of S, radomly ad idepedetly, with probability p. The expected umber of copies of T i S is pµ, ad the expected umber of pairs of members of S that share a edge is 1 2 p2. By omittig a arbitrarily chose member of each such pair, we obtai a collectio of pairwise edgedisjoit copies of T whose expected umber is at least pµ 1 2 p2. Thus E(X) pµ p2 2 N N 4 4 (2 + o(1)) 88 N 2 = (1 4 + o(1))n2 4, 10
11 establishig (5). To apply Talagrad s Iequality (i the form preseted, for example, i [AS00], Chapter 7), ote that h(t ) is a Lipschitz fuctio, that is h(t ) h(t ) 1 if T, T differ i the orietatio of at most oe edge. Note also that h is fcertifiable for f(s) = ( 2) s. That is, wheever h(t ) s there is a set of at most ( 2) s orieted edges of T such that for every ordered touramet T which agrees with T o these edges, we have h(t ) s. By Talalgrad s Iequality we coclude that for every b ad t P r[x b t f(b)] P r[x b] e t2 /4. (6) Let B deote the media of X = h(t ). Without tryig to optimize the absolute costats, we claim that B N (7) Ideed, assume this is false, ad apply (6) with b = N obtai that P r[x N N ] P r[x N 2 ad t = N. As f(b) = ( ] e N 2 /(64 6). ) N 2 N , we Sice, by assumptio, B < N , the first term of the lefthad side is at least 1/2, ad we coclude that P r[x N ] 2e N 2 /(64 6). As X = h(t ) ( N ) for every T, this implies that E(X) N 2 ( ) N e N 2 /(64 6) = N o(1), cotradictig (5) ad hece provig (7). We ca ow apply (6) with b = N ad t = N 4 3 to obtai that P r[x = 0]P r[x b] P r[x b t f(b)]p r[x b] e N 2 /(64 6). By (7), we have P r[x b] 1/2, ad hece P r[x = 0] 2e N 2 /(64 6). Thus, we have proved that, for every fixed orderig of T ad for every fixed ordered T, the probability that T cotais o ordered copy of T is at most 2e N 2 /(64 6). Sice the total umber of orderigs of T is less tha N N = e N log N ad the total umber of touramets T o vertices is 2 ( 2) we coclude that the probability that (T, < ) fails to cotai some touramet of size i some orderig is at most e N log N 2 ( 2) 2e N 2 /(64 6) = o(1). 11
12 This completes the proof. The above proof ca be modified to deal with graphs i the place of touramets. We obtai the followig, which is a very slight umerical improvemet of the mai result i [BK93]. Theorem 3.6 (see also [BK93]) Give a iteger N, let 0 be the largest iteger such that ( ) N 2 ( 0 2 ) 1, 0 ad put = 0 2. The, for all sufficietly large N, the followig holds almost surely. The radom graph G(N, 1/2) cotais, i ay orderig, a iduced copy of every ordered graph o at most vertices. 4 Touramets ad Hfree graphs I this sectio, we prove Theorem 1.2. We eed the followig wellkow Lemma 4.1 ([ES35]) For ay two total orderigs of the same (k 2 + 1)elemet set V, there is a (k + 1)elemet subset U V such that either the order of ay two elemets of U is the same, or the order of ay two elemets is opposite i the two orderigs. We say that a touramet T has the ErdősHajal property if there exists a positive ε = ε(t ) such that every T free touramet with vertices has a trasitive subtouramet whose size is at least ε. To ay touramet T ad to ay orderig < of its vertex set, assig a ordered graph (H(T ), <) o the same vertex set, as follows. Joi two vertices u < v by a edge of H(T ) if ad oly if the edge coectig them i T was directed towards v. Similarly, assig to ay ordered graph (H, <) a ordered touramet (T (H), <) with the same vertex set, by coectig u < v with a edge directed towards v if uv E(H) ad with a edge directed towards u if uv / E(H). Now we have everythig eeded for the Proof of Theorem 1.2: Assume first that Cojecture 1 is true, i.e., every graph has the Erdős Hajal property. Let T be a touramet. We wat to show that T also has the ErdősHajal property. Choose a arbitrary orderig < of the vertex set of T. Applyig Theorem 3.4 to the ordered graph (H(T ), <) associated with T ad <, we obtai that there exists a graph H with the property that, for ay orderig < of H, (H(T ), <) is a iduced subgraph of (H, < ). By Cojecture 1, there exists a ε(h ) > 0 such that every H free graph with vertices has a homogeeous subset of size at least ε(h ). 12
13 Cosider ow a T free touramet T with vertices ad a orderig < of V (T ). The the ordered graph (H(T ), < ) associated with them caot cotai a iduced subgraph isomorphic to H (because, o matter how it is ordered, this would yield a copy of T i T ). Thus, H(T ) must have a homogeeous set of size at least ε(h ). However, a homogeeous set i (H(T ), < ) correspods to a trasitive subtouramet i T. The proof of the reverse statemet is very similar, but the roles of graphs ad touramets have to be switched. Assume that Cojecture 2 is true, ad let H be a arbitrary graph. To establish that H has the ErdősHajal property, fix a liear order < o V (H), ad deote the associated ordered touramet by (T (H), <). By Theorem 1.3, there exists a touramet T with the property that, for ay orderig < of T, (T (H), <) is a subtouramet of (T, < ). By Cojecture 2, there exists a ε(t ) > 0 such that every T free touramet with vertices has a trasitive subtouramet of size at least ε(t ). Cosider ow a Hfree graph H with vertices ad a orderig < of V (H ). The the ordered touramet (T (H ), < ) associated with them caot cotai a subtouramet isomorphic to T (because, o matter how it is ordered, this would yield a copy of H i H ). Thus, T (H ) must have a trasitive subtouramet of size at least ε(h ). However, by Lemma 4.1, ay such subtouramet has at least ε(h )/2 vertices such that, with respect to the orderig <, either all edges coectig them are directed towards their larger edpoits, or all of them are directed towards their smaller edpoits. These vertices iduce a complete or a empty subgraph of H, respectively. Ackowledgmet We would like to thak two aoymous referees for helpful commets. Refereces [AS00] N. Alo ad J. H. Specer: The Probabilistic Method, Secod Editio, Wiley, [A68] B. Alspach: A combiatorial proof of a cojecture of Goldberg ad Moo, Caad. Math. Bull. 11 (1968), [BK93] G. Brightwell ad Y. Kohayakawa: Ramsey properties of orietatios of graphs, Radom Structures ad Algorithms 4 (1993), [D67] J. D. Dixo: The maximum order of the group of a touramet, Caad. Math. Bull. 10 (1967), [EEH73] R. C. Etriger, P. Erdős, ad C. C. Harer: Some extremal properties cocerig trasitivity i graphs, Periodica Mathematica Hugarica 3 (1973), [E47] P. Erdős: Some remarks o the theory of graphs, Bulleti of the America Mathematical Society 53 (1947),
14 [EH89] P. Erdős ad A. Hajal: Ramseytype theorems, Discrete Mathematics 25 (1989), [ES35] P. Erdős ad G. Szekeres: A combiatorial problem i geometry, Compositio Math. 2 (1935), [G97] A. Gyárfás: Reflectios o a problem of Erdős ad Hajal, i: The Mathematics of Paul Erdős (R. L. Graham ad J. Nešetřil, eds.), Algorithms ad Combiatorics 14, Volume II, SprigerVerlag, Heidelberg, 1997, [K95] N. N. Kuzjuri: O the differece betwee asymptotically good packigs ad coverigs, Europea J. Comb. 16 (1995), [L72] L. Lovász: Normal hypergraphs ad the perfect graph cojecture, Discrete Mathematics 2 (1972), [L83] L. Lovász: Perfect Graphs, i: Selected Topics i Graph Theory, Volume 2 (L. W. Beieke, R. J. Wilso, eds.), Academic Prcess, LodoNew York, 1983, [RW89] V. Rödl ad P. Wikler: A Ramseytype theorem for orderigs of a graph, SIAM Joural of Discrete Mathematics 2 (1989), [Se74] D. Seische: O a property of the class of colorable graphs, Joural of Combiatorial Theory, Ser. B 16 (1974), [S74] J. Specer: Radom regular touramets, Periodica Mathematica Hugarica 5 (1974), [T95] M. Talagrad, Cocetratio of measure ad isoperimetric iequalities i product spaces, Ist. Hautes Études Sci. Publ. Math. 81 (1995)
Department of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationMATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More information0,1 is an accumulation
Sectio 5.4 1 Accumulatio Poits Sectio 5.4 BolzaoWeierstrass ad HeieBorel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationrepresented by 4! different arrangements of boxes, divide by 4! to get ways
Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationJournal of Combinatorial Theory, Series A
Joural of Combiatorial Theory, Series A 118 011 319 345 Cotets lists available at ScieceDirect Joural of Combiatorial Theory, Series A www.elsevier.com/locate/jcta Geeratig all subsets of a fiite set with
More informationON THE EDGEBANDWIDTH OF GRAPH PRODUCTS
ON THE EDGEBANDWIDTH OF GRAPH PRODUCTS JÓZSEF BALOGH, DHRUV MUBAYI, AND ANDRÁS PLUHÁR Abstract The edgebadwidth of a graph G is the badwidth of the lie graph of G We show asymptotically tight bouds o
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationAdvanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σalgebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More information1.3. VERTEX DEGREES & COUNTING
35 Chapter 1: Fudametal Cocepts Sectio 1.3: Vertex Degrees ad Coutig 36 its eighbor o P. Note that P has at least three vertices. If G x v is coected, let y = v. Otherwise, a compoet cut off from P x v
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationAN ASYMPTOTIC ROBIN INEQUALITY. Patrick Solé CNRS/LAGA, Université Paris 8, SaintDenis, France.
#A8 INTEGERS 6 (206) AN ASYMPTOTIC ROBIN INEQUALITY Patrick Solé CNRS/LAGA, Uiversité Paris 8, SaitDeis, Frace. sole@est.fr Yuyag Zhu Departmet of Math ad Physics, Hefei Uiversity, Hefei, Chia zhuyy@hfuu.edu.c
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationPerfect Packing Theorems and the AverageCase Behavior of Optimal and Online Bin Packing
SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the AverageCase Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More information4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.
4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationOn the L p conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/0302376, ublished olie 2007080 DOI 0.007/s0003007993x Archiv der Mathematik O the L cojecture for locally comact
More informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More informationApproximating the Sum of a Convergent Series
Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationπ d i (b i z) (n 1)π )... sin(θ + )
SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationThe Diameter of Sparse Random Graphs
Advaces i Applied Mathematics 26, 257 279 (200 doi:0.006/aama.200.0720, available olie at http://www.idealibrary.com o The Diameter of Sparse Radom Graphs Fa Chug ad Liyua Lu Departmet of Mathematics,
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationSection 73 Estimating a Population. Requirements
Sectio 73 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationDIRECTED GRAPHS AND THE JACOBITRUDI IDENTITY
Ca. J. Math., Vol. XXXVII, No. 6, 1985, pp. 12011210 DIRECTED GRAPHS AND THE JACOBITRUDI IDENTITY I. P. GOULDEN 1. Itroductio. Let \a i L X deote the X determiat with (/', y)etry a, ad h k = h k (x
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationNonnegative ksums, fractional covers, and probability of small deviations
Noegative ksums, fractioal covers, ad probability of small deviatios Noga Alo Hao Huag Bey Sudakov Abstract More tha twety years ago, Maickam, Miklós, ad Sighi cojectured that for ay itegers, k satisfyig
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationA NONRECURSIVE ALGORITHM FOR POLYGON TRIANGULATION. Predrag S. STANIMIROVI], Predrag V. KRTOLICA, Rade STANOJEVI] 1. INTRODUCTION AND PRELIMINARIES
Yugoslav Joural of Operatios Research 13 (2003), Number 1, 6167 A NONRECURSIVE ALGORITHM FOR POLYGON TRIANGULATION Predrag S. STANIMIROVI], Predrag V. KRTOLICA, Rade STANOJEVI] Faculty of Sciece ad Mathematics,
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More information6 Borel sets in the light of analytic sets
Tel Aviv Uiversity, 2012 Measurability ad cotiuity 86 6 Borel sets i the light of aalytic sets 6a Separatio theorem................. 86 6b Borel bijectios.................... 87 6c A oborel aalytic set
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationSection 1.6: Proof by Mathematical Induction
Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems
More informationEconomics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is
More information2. Degree Sequences. 2.1 Degree Sequences
2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we
More information