ON SOME EXTREMUM PROBLEMS IN ELEMENTARY GEOMETRY


 Willis Walsh
 1 years ago
 Views:
Transcription
1 ON SOME EXTREMUM PROBLEMS IN ELEMENTARY GEOMETRY P. ERD& and G. SZEKERES* Mathematical Institute, EBtVBs Lotand University, Budapest and University Adelaide (Received May 20, 7960) Dedicated to the memory of Professor L, FEJ~R 1. Let S denote a set of points in the plane, N(S) the number of points in S. More than 25 years ago we have proved [2] the following conjecture of ESTHER KLEIN SZERERES: There exists a positive integer f (n) with the property that if iv (S) > f(n) then S contains a subset P with N(P) = n such that the points of P form a convex ngon. Moreover we have shown that if lo(n) is the smallest such integer then fcl(n) 5 i n g and conjectured that fo(n) = 2n2 for every n 2 3.l We are unable to prove or disprove this conjecture, but in $ 2 we shall construct a set of 2n2 points which contains no convex ngon. Thus 2*f! If0 {n)< 2rz. I ~ n2 1 A second problem which we shall consider is the following: It was proved by SZEKERES [3] that (i) In any configuration of iv = 2n + 1 points in the plane there are three points which form an angle (I z) greater than (1  l/n + I/n NZ) X. (ii) There exist configurations of 2n points in the plane such that each angle formed by these points is less than (I  I/n) z + E with E > 0 arbitrarily small. The first statement shows that for sufficiently small E > 0 there are no configurations of points which would have the property (ii). Hence in a certain sense this is a best possible result; but it does not determine the exact limiting value of the maximum angle for any given N(S). Let C( (m) denote the greatest positive number with the property that hr every configuration of m points in the plane there is an angle,9 with (1) f3 "u(m). * This paper was written while P. ERDGS was visiting at the University of Adelaide. I The conjecture is trivial for n = 3; it was proved by Miss KLEIN? for n = 4 and by E. MAKAI and P. TURAN for n = 5.
2 54 P. ERDOS AND G. SZEKERES From (i) and (ii) above it follows that CI (~1) exists for every uz 2 3 and that for 2n < m I 2 +1, (2) [ $ l/n (2n + I) ] $I I CL (m) I [ 1  l/(n + l)] z. Two questions arise in this connection: 1. What is the exact value of Q (m), 2. Can the inequality (1) be replaced by (3) P > a (ml For the first few values of m one can easily verify that a (3) = + a (4) = in, CI (5) = $76, a (6) = a (7) = ct (8) = + zr, and that the strict inequality (3) is true for m = 7 and 8. For 3 I m I 6 the regular uz gons represent configurations in which the maximum angle is equal to CL (m); but we know of no other cases in which the equality sign would be necessary in (1). In $ 4 we shalt prove THEOREM 1. Every plane configuration of 2 poinfs cn 2 3) contains an angr e greater than (11 in) S. The theorem shows in conjunction with (ii) above that for n r 3, CL (29 = = (1  I/n) z and that the strict inequality (3) holds for these values of m. The problem is thus completely settled for m = 2, n 2 2. It is not impossible that cx (m) = (1  l/n) YC for 2 l< m < 2, n 2 4, and that (3) holds for every m > 6. However, we can only prove that for 0 ( k < 2 l, cy. (an  k) 2 (11 /n) YE  k x/2 (an k) (Theorem 2). Finally we mention the following conjecture of P. ERDBS : Given 2n+ 1 points in nspace, there is an angle determined by these points which is greater than 1  X. For n = 2 the statement is trivial, for n = 3 it was proved by N. H. 2 KUIPER nad A. II. BOERDIJK. For n > 3 the answer is not known; and it seems that the method of 9 4 is not applicable to this problem. ** 2. In this section we construct a set of 2n2 points in the plane which contains no convex ngon. For representation we use the Cartesian (x, y) plane. All sets to be considered are such that no three points of the set are collinear. A sequence of points!x, V ), v = 091, * , k is said to be convex, of length k, if x, < x, <... < Xfi Yv  yv1 ( YYtcYY for v=l,.... kl; x,jx,,_, xtl+lx,, 2 Unpublished. ** (Added in proof: This conjecture was recently proved by DANZER and GR~~NBAUM in a surprisingly simplex way.)
3 ON SOME EXTREMUM PROBLEMS 57 Suppose now that P,, (i = 1,..., T) is a nonempty subset of Ski, 1 I k, < <...<k,<n 1 and such that P = 6 Pi forms a convex polygon. Since i=l the slope of lines within each SI,, is positive, Pi for 1 < i < r consists of a single point and P, must form a concave sequence, P, a convex sequence. But then the total number of points in P is at most k, + (k,  k,  1) + (nkk,) = n The proof of Theorem 1 requires a refinement of the method used in [3], and in this section we set up the necessary graphtheoretical apparatus. We denote by C.(N) the complete graph of order N, i. e. a graph with N vertices in which any two vertices are joined by an edge. Jf G is a graph, then S (G) shall denote the set of vertices of G. If A is a subset of S (G), then G]A denotes the restriction of G to A. An even (odd) circuit of G is a closed circuit containing an even (odd) number of edges. A parfifion of G is a decomposition G = G, G, into subgraphs Gi with the following property: Each Gi consists of all vertices and some edges of G such that each edge of G appears in one and only one Gi (Gi may not contain any edge at all). We call a partition G = G G, even if it has the property that no Gi contains an odd circuit. LEMMA 1. If a graph G contains no odd circuit fken its vertices can be divided in two classes, A and B, suck that every edge of G has one endpoint in A and one in B. This is wellknown and very simple to prove, e. g. [I], p 170. LEMMA 2. If C(N) = G G, is an even parfition of the complefe graph ON) into n parts then N _< 2. This Lemma was proved in 131; for the sake of completeness we repeat the argument. Since G1 contains no odd circuit we can divide S (C(N)) in classes A and B, containing N, and N, vertices respectively, such that each edge of G1 connects a point of A with a point of B. But then G, G, induces an even partition Gi +... I GA of G = ON) j A and since G is a complete graph of ordtr N,, we conclude by induction that N, I 2+l. Similarly N, I 2 l hence N = N, $ N2 K 2. To prove Theorem 1 we shall need more precise information about the structure of even partitions of CY ), particularly in the limiting case of N = 2. The following Lemmas have some interest of their own; they are formulated in greater generality than actually needed for our present purpose. LEMMA 3. Let N = 2 and UN) = G G, an even parfition o/ UN) into n parts. Then the total number of edges emanating from a fixed vertex p E S (UN)) in Gi, Gjj, where 1 < jl < j2 <... < ji I n, is at least 21 and at most Clearly the order in which the components Gi are written is immaterial, therefore we can assume in the proof that jv = Y, v = 1,.., i. We also note that the first statement follows from the second one by applying the latter to the complementary partition G,+l G, and by noting that the number of edges from p in G, +... A G,, is 21. We shall prove the second state
4 58 P. ERDiiS AND G. SZEKERES ment in the form that there are at least 2  vertices in S (Cc )) (including y itself) which are not joined with p in G, Gi. The statement is trivial for n = 1; we may therefore assume the Lemma for n  1. By assumption, G, contains no odd circuit. Therefore by Lemma 1 we can divide its vertices into two classes, A and B, such that no two points of A (or of B) are joined in G,. Both classes A and B contain 2  vertices; for otherwise one of them, say A, would contain more than 2 l vertices and CYN)J A would have an even partition Gh +.., J GA into n  1 parts, contrary to Lemma 2. Let A be the class containing p. Hence there are at least 2 l vertices with which p is not joined in G,. This proves the Lemma for i = 1. Suppose i > 1 and consider the partition GA +.,. + GL of UN)1 A, induced by G, $ G, G,. Since the order of UN)1 A is 2 l, we find by the induction hypothesis that there are at least 2n1([i) = 2  vertices in A to which p is not joined in Gi G:. But p is not joined with any vertex of A in G,, therefore it is not joined with at least 2nl vertices in G, +... f G,. In the special case of i = 1 we obtain LEMMA 3.1. Let N = 2n and ) = Gi G, an even partition. Then every p is an endpoint of at least one edge in every Gi. If N < 2n, then Lemma 3.1 is no longer true, but the number of vertices for which it fails cannot exceed 2  N. More precisely we shall prove LEMMA 4. Let N = 2 k, 0 I k < 2, and UN) = G, G, an even partition of UN) into n parts. Denote by Y(P), p E S = S (UN)) the number of graphs Gi in which there is no edge from p. Then ip (29(P)  1) I k. PCS PROOF. For n = 1 the statement is trivial, therefore assume the Lemma for Let ql,..., qj be the exceptional vertices in G, from which there are no edges in G, and denote by Q the union of vertices qi, i = 1,...,. (Q may be empty). By Lemma 1, S is the union of disjoint subsets A, B and 6 such that every edge in G, has one endpoint in A and one in B. Denote by A, the union of A and Q, by B, the union of B and Q. Let a and b be the number of vertices in A and B respectively. Then a + b + i = 2n  k and a + i I 2n1, b + i I 2 l by Lemma 2, applied to UN) IA, and CcN)J Bl. Write a = 2 l  i k,, b = 2+l  j  k, so that k, 2 0, k, 2 0 and 2n  i  k,  k2 = = 2 k 7 (4 k = j + k, + kz. By applying the induction hypothesis to W)JA, and to the partition G;$... + G,: induced by G G,, we find 2 (21*(P)  1) + 2 1) I; k, PEA P?Q for some p(p) 2 v (p). Therefore a fortiori F (2(P) I)+ 2 (2v(Pl11)(x.1 p% PCQ
5 ON SOME EXTREMUM PROBLEMS and similarly Hence 7 (2*(p) 1)  j<k,+k,=kj PT.. by (4), which proves the Lemma. 4. Before proving Theorem 1 we introduce some further notations and definitions. To represent points in the Euclidean plane E we shall sometimes use the complex plane which will also be denoted by E. If ql, p, q2 are points in E, not on one line and in counterclockwise orientation, the angle (< z) formed by the lines pql and pql will be denoted by A (ql p q2). A set of points S in E is said to have the property P, if the angle formed by any three points of S is not greater than (1  l/n) X. We shall briefly say that S is P, or not P, according as it has or has not this property. A direction ti in E is a vector from 0 to elz on the unit circle. An npartition of E with respect to the direction u is a decomposition of E  (0) into sectors T,>, k = 1,...? 2n where T,< consists of all points z = yei(e+o), r > 0, (k  1) 2 < g?<k 2. n n With every set of points S = {pl,..., pn} and every rzpartition of E with respect to some direction w we associate a partition UN) = G, G, of UN) in II parts according to the following rule: pp, py are joined in G, if and only if the vector from p,, to py is in one of the sectors Ti, T,+i. The following Lemma was proved in [3]. LEMMA 5. If the set S is P, then the partition UN) = G, G, associated with any given npartition of E is necessarily even. We shall also need LEMMA 6. If p1 pz... p, are consecutive vertices of a regular ngorz P and q is a point distinct from the centre and inside P then there is a pair of vertices (P,, PJ such that A (p,q Pj) > (1  l/n) n. The proof is quite elementary; if pl is a vertex nearest to q and if q is in the triangle pi pj P,+~ then at least one of the angles A (p, q p,), A (p, q pi1) is >( 1l/n)niT3 Lemma 5 and Lemma 2 give immediately the result that a set of points in the plane cannot be P,. Our purpose, however, is to prove Theorem 1 which can be stated as follows: THEOREM le. A set of 2n points in the plane is not P,. PROOF. Let S be a set of iv = 2 (n > 2) points in the plane, pl, pz,..., pi; the vertices of the least convex polygon of S, in cyclic order and counterclockwise orientation. We can assume that no three vertices of S are collinear, otherwise S is obviously not P,. We distinguish several cases. 3 See Probleln 4056, Amer. Mafh. A4onthl~, 54 (1947), p Solution by C. R. PHELPS.
6 P. ERDOS AND G. SZEKERES Suppose first that there is an angle A (pip1 pi pii1) < (1  l/n) r. Let 01 be the direction pi pi+1 and u (r~) the npartition of E with respect to c(. Then there are no points of S in the sectors T, and Tn.+1 corresponding to a(~), hence there is no edge from the vertex pi in the component G, of the associated partition CtN) = G, I G,. We conclude from Lemma 3.1 that G A G, is not an even partition, hence by Lemma 5, S is not P,. Henceforth we assume that all angles A (pie1 pi JI~+~) are equal to (1l in) x so that the least convex polygon has 2n vertices. For convenience let these vertices be (in cyclic order and counterclockwise orientation) pi q1 p2 qz... pn qn, and denote by ai the side length pi qi and by bj the side qi P~.+.~. Suppose that all angles A (piipipi+l) and A (Qi1qiQi+l) are equal to (12/n) z. Then an elementary argument shows that the triangles pidl Qi1 pi, qi1 piqi, prqipitl etc are similar, hence a,1 b,+, _ uj =  =... ai bi (Ii+1 which implies a, = a, =... = a,, b, = b, =... = b,. We conclude that Pl PL.. * pn is a regular ngon. Now let q be any point of S inside the least convex polygon. If q is inside the triangle pi qi pi+l then clearly A(p, q P!+~) > (1  l/n) R. Therefore we may assume that each q inside the least convex polygon is already inside p1,.. pn. Since n 2 3, there are at least two such points, hence we may assume that q is not the centre. But then by Lemma 6, A (pi q pj) > (1  l/n) n for suitable i and j. The last remaining case to be considered is when not all angles A (pi1pi P~+~) are equal; then for some i, A (pi1 pi P~+~) < (12/n) it. Since A (Pi1 qi1 Pi> = A (Pi qi Pi+d = (11/ 12 ) E, we may assume that there are no points of S inside the triangles pil qia1 pi and pi qi P+~. But then if cx is the direction of pi pi+l and r (CZ) the corresponding npartition of E, UN) = G1+... jg,, the partition of UN) associated with CT (a), then in GnF1 + G, there are only two edges running from pi, namely pi qil and pi qi. By Lemma 3 (with i = 2, jl = n  1, jz = n), UN) = G G, is not an even partition and by Lemma 5, S is not P,. Finally we prove THEOREM 2. In a plane configurafion of N = 2n  there is an angle 2 (11 /n k/2 N) TL k points (0 < k < 2 l) PROOF. Suppose that all angles formed by the points of S are I (11 in) x  f a 6 where 6 = kr/n and 6 >0. Let pe S and y=q(p)=a(q,pq,) I (ll/n)a+y,q,es, qzes the maximum angle at p. If there are several such angles, make an arbitrary but fixed choice.
7 ON SOME EXTREMUM PROBLEMS 61 Let u = CL (p) be the direction of p q1 so that there are no lines pi, q E S in the sectors If S has N = 2  k points, then there are N pairs of directions 4 ~1 (p). Hence there is a such that at least k + 1 directions E,CX (p.), v = 0, 1,.. *, k, F, = 1 1 are in the interval By a slight displacement of p we can achieve that all directions in S should be different from the direction p Consider now the npartition u /? of E with respect to,8 + p 6 I 4i and construct the associated partition ON) = G, G, with the following modification: Every p q with direction between p + $SandB+ S++6 shall be joined in G, instead of G, and every p q with direction between j3 +f n shall be joined in Gz instead of G,. With this modification 4 it is still true tha: the partition UN) = G1 +. a, + G, is even if S has no angle greater or equal to (11 /n) zf 68. But it is easy to see that the vertices p,. are not joined with any vertex of UN) in G,. For if p + <p then the edges emanating fromp, in the sectors T,, Tnfl are count ed to G,; and if p < E, CL (pp) < p + $ 6, the only possible edges from p,, in the sectors T,, T,+l are in directions between p + $ S+Eandj3+ $3+z n n and these are counted to G?. Thus we have a contradiction with Lemma 4 and the Theorem is proved. NOTE ADDED IN PROOF. In the case of k = 1, Theorem 2 can be improved as follows. THEOREM 3. Every plane configuration of 21 points (n 2 2) contains an angle not less than (1  l/n) X. The theorem shows in particular that CI (2N 1) = (11 /n) X, but we cannot decide whether the strict inequality (3) is valid for m = 21.
8 62 P. ERDCiS AND G. SZEKERES: ON SOME EXTREMUM PROBLEMS PROOF. Suppose N (S) = 2n  1 and all angles in S are less than (I 1 In) z. Let q be an interior point of S, that is one which is not on the least convex polygon. Let A (41 q 41) = ( 11 in) x  6, a>0 be the largest angle at q. If p is the direction of qql, a =,8 + i 6, CT (a) the corresponding npartition of E, UN) = G, G, the partition of Oh ) associated with LT (a), then clearly there are no edges from q in G1. We conclude from Lemma 4 and Lemma 5 that from all other vertices of S there is at least one edge in every Gi (i = 1,..., n). We show that this leads to a contradiction. LetP=(p,,.... pm) denote the least convex polygon of S. Since each angle in P is less than (1  I /II) z, we have m I 2 n  1. Therefore if T,,..., T,, are the sectors of CT (a) there is a p, such that if piipi is in T,, then pi plil is not in T,<. But then there is no edge from pi in Gk, as easily seen by elementary geometry. References [I ] K~NIG, D., Theorie der endlichen und unendlichen Graphen, (Leipzig, 1536). (21 ERDBS, I?. and SZEKERES, G., A combinatorial problem in geometry, Composifio Mafh., 2 (1935), [3] SZEKERES, G., On an extremmn problem in the plane, Amer. Journal of Math., 63 (1Q41), 2CB210.
Graphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More informationERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS*
BOLYAI SOCIETY MATHEMATICAL STUDIES, 11 Paul Erdos and his Mathematics. II, Budapest, 2002, pp. 367373. ERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS* Problems and results
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationLattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College
Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationThe number of generalized balanced lines
The number of generalized balanced lines David Orden Pedro Ramos Gelasio Salazar Abstract Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ 0. A line l
More informationOdd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A longstanding
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction
ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationProblem Set I: Preferences, W.A.R.P., consumer choice
Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y
More informationPROBLEM SET 7: PIGEON HOLE PRINCIPLE
PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics
More informationTAKEAWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION
TAKEAWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf takeaway?f have become popular. The purpose of this paper is to determine the
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationR u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c
R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 22010, January 24, 2010 RUTCOR Rutgers Center for Operations
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationStudia Scientiarum Mathematicarum Hungarica 41 (2), 243 266 (2004)
Studia Scientiarum Mathematicarum Hungarica 4 (), 43 66 (004) PLANAR POINT SETS WITH A SMALL NUMBER OF EMPTY CONVEX POLYGONS I. BÁRÁNY and P. VALTR Communicated by G. Fejes Tóth Abstract A subset A of
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationHOLES 5.1. INTRODUCTION
HOLES 5.1. INTRODUCTION One of the major open problems in the field of art gallery theorems is to establish a theorem for polygons with holes. A polygon with holes is a polygon P enclosing several other
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationDivisor graphs have arbitrary order and size
Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More informationExponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A klabeling of vertices of a graph G(V, E) is a function V [k].
More informationx 2 f 2 e 1 e 4 e 3 e 2 q f 4 x 4 f 3 x 3
KarlFranzensUniversitat Graz & Technische Universitat Graz SPEZIALFORSCHUNGSBEREICH F 003 OPTIMIERUNG und KONTROLLE Projektbereich DISKRETE OPTIMIERUNG O. Aichholzer F. Aurenhammer R. Hainz New Results
More informationThe chromatic spectrum of mixed hypergraphs
The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).
Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationG = G 0 > G 1 > > G k = {e}
Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationarxiv:1203.1525v1 [math.co] 7 Mar 2012
Constructing subset partition graphs with strong adjacency and endpoint count properties Nicolai Hähnle haehnle@math.tuberlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationOn the crossing number of K m,n
On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known
More informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More informationLecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationA Sublinear Bipartiteness Tester for Bounded Degree Graphs
A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublineartime algorithm for testing whether a bounded degree graph is bipartite
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationWhen is a graph planar?
When is a graph planar? Theorem(Euler, 1758) If a plane multigraph G with k components has n vertices, e edges, and f faces, then n e + f = 1 + k. Corollary If G is a simple, planar graph with n(g) 3,
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationResource Allocation with Time Intervals
Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationCycles and cliqueminors in expanders
Cycles and cliqueminors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor
More informationA Hajós type result on factoring finite abelian groups by subsets II
Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is
More informationeach college c i C has a capacity q i  the maximum number of students it will admit
n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i  the maximum number of students it will admit each college c i has a strict order i
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationarxiv: v1 [math.ho] 7 Apr 2016
ON EXISTENCE OF A TRIANGLE WITH PRESCRIBED BISECTOR LENGTHS S. F. OSINKIN arxiv:1604.03794v1 [math.ho] 7 Apr 2016 Abstract. We suggest a geometric visualization of the process of constructing a triangle
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More information