ë2ë P. Erdíos, Problems and results in additive number theory, in Colloque ë4ë P. Erdíos, Graph Theory and Probability II., Canad. J. Math.

Transcription

1 ëë P. Erdíos, Probles ad results i additive uber theory, i Colloque sur la Thçeorie des Nobres ècbrmè, Bruxelles, 1955, ë3ë P. Erdíos, Graph Theory ad Probability, Caad. J. Math. 11 è1959è, ë4ë P. Erdíos, Graph Theory ad Probability II., Caad. J. Math. 13 è1961è, ë5ë P. Erdíos, O circuits ad subgraphs of chroatic graphs, Matheatika 9è196è, ë6ë P. Erdíos, O a cobiatorial proble I., Nordisk. Mat. Tidskr. 11 è1963è, ë7ë P. Erdíos, O a cobiatorial proble II., Acta. Math. Acad. Sci. Hugar. 15 è1964è, ë8ë P. Erdíos ad J. W. Moo, O sets of cosistet arcs i a touraet, Caad. Math. Bull. 8 è1965è, ë9ë P. Erdíos ad A. Rçeyi, O the evolutio of rado graphs, Mat. Kutatço It. Kíozl. 5 è1960è, ë10ë P. Erdíos ad J. Specer, Ibalaces i k-coloratios, Networks 1 è197è, ë11ë P. Erdíos ad G. Szekeres, A cobiatorial proble i geoetry, Copositio Math. è1935è, ë1ë W. F. de la Vega, O the axial cardiality of a cosistet set of arcs i a rado touraet, J. Cobiatorial Theory, Series B 35 è1983è,

2 7 1965: Urakable Touraets Let T be a touraet with players 1;...; èeach pair play oe gae ad there are o tiesè ad ç a rakig of the players, techically a perutatio o f1;...;g. Call gae fi; jg a oupset if i beats j ad çèiè éçèjè; a upset if i beats j but çèjè é çèiè. The æt f èt;çè is the uber of oupsets ius the uber of upsets. Oe ight have thought - i preprobabilistic days! - that every touraet T had a rakig ç with a reasoably good æt. With J.W. Moo, Erdíos ë?ë easily destroyed that cojecture. Theore. There is a T so that for all ç fèt;çè ç 3= èl è 1= Thus, for exaple, there are touraets so that uder ay rakig at least 49è of the gaes are upsets. Erdíos ad Moo take the rado touraet, for each pair fi,jg oe ëæips a fair coi" to see who wis the gae. For ay æxed ç each gae is equally likely to be upset or oupset, ad the diæeret gaes are idepedet. Thus fèt;çè ç S, where = ad S is the uber of heads ius the uber of tails i æips of a fair coi. Large deviatio theory gives PrëS éæë ée, æ Oe ow uses very large deviatios. Set æ = 3= èl è 1= so that the above probability is less tha, é!,1. This supersall probability is used because there are! possible ç. Now with positive probability o ç have fèt;çè éæ.thus there is a T with o ç havig fèt;çè éæ. The use of extree large deviatios has becoe a aistay of the Probabilistic Method. But I have a ore persoal reaso for cocludig with this exaple. Let gèè be the least iteger so that every touraet T o players has a rakig ç with fèt;çè ç gèè. The gèè ç 3= èl è 1=. Erdíos ad Moo showed gèè éc, leavig ope the asyptotics of gèè. I y doctoral dissertatio I showed gèè éc 1 3= ad later èbut see delavega ë?ë for the ëbook proof" è that gèè éc 3=. Though at the tie I was but a æ Paul respoded with his characteristic opeess ad soo ë?ë I had a Erdíos uber of oe. Thigs have't bee the sae sice. Refereces ë1ë P. Erdíos, Soe rearks o the theory of graphs, Bull. Aer. Math. Soc. 53 è1947è,

3 Proof: Color æ radoly. Each A i has probability 1, of beig oochroatic, the probability soe A i is oochroatic is the at ost 1, é 1 so with positive probability oa i is oochroatic. Take that colorig. I 1964 Erdíos ë?ëshowed this result was close to best possible. Theore. There exists a faily A with = c which is ot -colorable. Here Erdíos turs the origial probability arguet iside out. Before the sets were æxed ad the colorig was rado, ow, essetially, the colorig is æxed ad the sets are rado. He sets æ = f1;...;ug with u a paraeter to be optiized later. Let A 1 ;...;A be rado -sets of æ. Fix a colorig ç with a red poits ad b = u, a blue poits. As A i is rado PrëçèA i è costatë =, a, u æ, + b æ æ, æ u= ç, u æ, x The secod iequality, which follows fro the covexity of æ, idicates that it is the equicolorigs that are the ost troublesoe. As the A i are idepedet Prëo A i oochroaticë ç "1, æ u=,, u æ è Now suppose u ", æ u= 1,, u æ è é 1 The expected uber of ç with o A i oochroatic is less tha oe. Therefore there is a choice of A 1 ;...;A for which o such ç exists, i.e., A is ot -colorable. Solvig, oe ay take l = d çe, l ç1, èu= è èè u, uæ, Estiatig, lè1, æè ç æ this is roughly cu = u= æ. This leads to a iterestig calculatio proble èas do ay probles ivolvig the Probabilistic Method!è í æd u so as to. The aswer turs out to be u ç = at which value ç èe l è,. Erdíos has deæed èè as the least for which there is a faily of - sets which caot be -colored. His results give æè è=èè =Oè è. Beck has iproved the lower boud to æè 1=3 è but the actual asyptotics of èè reai elusive. 9

4 cojecture that if every, say, =èl è vertices could be 3-colored the G could be 4-colored. This theore disproves that cojecture. We exaie the rado graph G ç Gè; pè with p = c=. Asithe 1957 paper PrëæèGè ç xë é è!è1, pè èxè h i x é èe=xèe,pèx,1è= x Whe c is large ad, say, x =10èl cè=c, the bracketed quatity is less tha oe so the etire quatity isoè1è ad a.s. æègè ç x ad so çègè ç c=è10 l cè. Give k Erdíos ay ow siply select c so that, with p = c=, çègè éka.s. Now for the local colorig. If soe set of ç æ vertices caot be 3- colored the there is a iial such set S with, say, jsj = i ç æ. I the restrictio Gj S every vertex v ust have degree at least 3 - otherwise oe could 3-color S,fvg by iiality ad the color v diæeretly fro its eighbors. Thus Gj S has at least 3i= edges. The probability ofg havig such as is bouded by!è æx, i! " æx e è i p 3i= ç ç ei 3 ç 3= ç c ç 3= è i 3i= i i=4 i=4, a, eployig the useful iequality bæ ç ea æ b. Pickig æ = æècè sall the b bracketed ter is always less tha oe, the etire su is oè1è, a.s. o such S exists, ad a.s. every æ vertices ay be 3-colored. Erdíos's ouetal study with Alfred Rçeyi ëo the Evolutio of Rado Graphs" ë?ë had bee copleted oly a few years before. The behavior of the basic graph fuctios such aschroatic ad clique uber were fairly well uderstood throughout the evolutio. The arguet for local colorig required a ëew idea" but the basic fraework was already i place è4: Colorig Hypergraphs Let A 1 ;...;A be -sets i a arbitrary uiverse æ. The faily A = fa 1 ;...; A g is -colorable èerdíos used the ter ëproperty B"è if there is a -colorig of the uderlyig poits æ so that o set A i is oochroatic. I 1963 Erdíos gave perhaps the quickest deostratio of the Probabilistic Method. Theoreë?ë: If é,1 the A is -colorable. 8

5 Each dètè has Bioial Distributio Bèx; pè ad so expectatio xp = æèl è so that oe ca get fairly easily EëZë =æè l è. Note this is the sae order as x. It is deæitely ot easy to show that for appropriate A; c èerdíos takes c = A,1= ad A largeè Zé 1, x with high probability. The requireet ëwith high probability" is quite severe. But ote, at least, that this is a pure probability stateet. Lets accept it ad ove o. Call a pair fi; jg çs soiled if it lies i a triagle with third vertex outside of S. At ost Z pairs are soiled so with high probability at least 1, x pairs are usoiled. Now we expose the edges of G iside S. If ay of the usoiled pairs are i G the G is good ad so the failure probability at ost è1, pè 1 è x è ée,æèpx è = o x!,1 1 A ad so G is good with high probability. Soud coplicated. Well, it is coplicated ad it is siultaeously a powerful applicatio of the Probabilistic Method ad a techical tour de force. The story has a coda: the Lovçasz Local Lea, developed i the id-1970s, gave a ew sieve ethod for showig that a set of bad evets could siultaeously ot hold. This author applied it to the rado graph Gè; pè with p = c,1= with the bad evets beig the existece of the various potetial triagles ad the idepedece of the various x-sets. The coditios of the Local Lea ade for soe calculatios but it was relatively straightforward to duplicate this result. Still, the ideas behid this proof, the subtle extesio of the Deletio Method otio, are too beautiful to be forgotte : No Local Colorig With his 1957 paper previously discussed Erdíos had already show that chroatic uber caot be cosidered siply a local pheoeo. With this result he puts the ail i the coæ. Theoreë?ë. For ay k ç 3 there is a æé0 so that the followig holds for all suæcietly large : There exists a graph G o vertices which caot be k-colored ad yet the restrictio of G to ay æ vertex subgraph ca be 3-colored. Ofte probabilistic theore are best uderstood as egative results, as couterexaples to atural cojectures. A priori, for exaple, oe ight 7

6 ever sice. We have already spoke of his 1947 paper o Rèk; kè. I his 1961 paper Erdíos ë?ë proves Rè3;kè éc k l k The upper boud Rè3;kè=Oèk èwas already apparet fro the origial Szekeres proof so the gap was relatively sall. Oly i 1994 was the correct order Rè3;kè=æè k l k è æally show. Erdíos shows that there is a graph o vertices with o triagle ad o idepedet set of size x where x = da 1= l e, ad A is a large absolute costat. This gives Rè3;xè éfro which the origial stateet follows easily. We'll igore A i our iforal discussio. He takes a rado graph Gè; pè with p = c,1=. The probability that soe x-set is idepedet is at ost è x! i x è1, pè xèx,1è= é he,pèx,1è= which isvery sall. Ufortuately this G will have lots èæè 3= èè of triagles. Oe eeds to reove a edge fro each triagle without akig ay of the x-sets idepedet. The Erdíos ethod ay be thought of algorithically. Order the edges e 1 ;...;e of G ç Gè; pè arbitrarily. Cosider the sequetially ad reject e i if it would ake a triagle with the edges previously accepted, otherwise accept e i. The graph G, so created is certaily triaglefree. What about the sets of x vertices. Call a set S of x vertices good èi G, ot G, èifit cotais a edge e which caot be exteded to a triagle with third vertex outside of S. Suppose S is good ad let e be such a edge. The S caot be idepedet ig,. If e is accepted we're clearly OK. The oly way e could be rejected is if e is part of a triagle e; e 1 ;e where the other edges have already bee accepted. But the e 1 ;e ust èas S is goodè lie i S ad agai S is ot idepedet. Call S bad if it is't good. Erdíos shows that alost always there are o bad S. Lets, say soethig occurs with high probability if its failure probability isoè xæ è,1 è. It suæces to show that a give S = f1;...;xg is good with high probability. This is the core of the arguet. We expose èto use oder teriologyè G i two phases. First we exaied the pairs fs; tg with s S; t 6 S. For each t 6 S let dètè be the uber of edges to S. Set è! dètè Z = X t6s 6

7 u was the X è, u!è, æ,, u! i=1 l, l u é è +1èè, è!u è,, éu, æ! è, æ é é è, æ è!u 1, u! é è,! æ!è, æ,, u! 1,, u,! é u e,u = Now the uber of possible choices for the u poits is è! é u éu u ad so the uber of graphs without the desired property is è,! èè, æ!! u 3 e,1+æ,ç = o as desired. Today, with large deviatio results assued beforehad, the proof ca be give i oe relatively leisurely page. May cosider this oe of the ost pleasig applicatios of the Probabilistic Method as the result sees ot to call for probability i the slightest ad earlier attepts had bee etirely costructive. The further use of large deviatios ad the itroductio of the Deletio Method greatly advaced the Probabilistic Method. Ad, ost iportat, the theore gives a iportat truth about graphs. I a rough sese the truth is a egative oe: chroatic uber caot be deteried by local cosideratios oly. é : Rasey Rè3;kè Rasey Theory was oe of Paul Erdíos's earliest iterests. The ivolveet ca be dated back to the witer of 193è33. Workig o a proble of Esther Klei, Erdíos proved his faous result that i every sequece of + 1 real ubers there is a ootoe subsequece of legth +1. At the sae tie, ad for the sae proble, George Szekeres rediscovered Rasey's Theore. Both arguets appeared i their 1935 joit paperë?ë. Bouds o the various Rasey fuctios, particularly the fuctio Rèl; kè, have fasciated Erdíos 5

8 uber ad arbitrarily high girth í i.e. o sall cycles. To ay graph theorists this seeed alost paradoxical. A graph with high girth would locally look like a tree ad trees ca easily be colored with two colors. What reaso could force such a graph to have high chroatic uber? As we'll see, there is a global reaso: çègè ç =æègè. To show çègè is large oe ëoly" has to show the oexistece of large idepedet sets. Erdíos ë?ë proved the existece of such graphs by probabilistic eas. Fix l; k, a graph is wated with çègè élad o cycles of size ç k. Fix æé 1 k, set p = æ,1 ad cosider G ç Gè; pè as!1. There are sall cycles, the expected uber of cycles of size ç k is kx i=3 èè i i pi = kx i=3 Oèèpè i è=oèè as kæ é 1. So alost surely the uber of edges i sall cycles is oèè. Also æx positive çéæ=. Set bu = 1,ç c. A set of u vertices will cotai, o average, ç ç u p= =æè æ è edges where æ =1+æ, ç é1. Further, the uber of such edges is give by a Bioial Distributio. Applyig large deviatio results, the probability of the u poits havig fewer tha half their expected uber of edges is e,cç.asæé1 this is saller tha expoetial, so oè, è so that alost surely every u poits have at least ç= edges. We eed oly that ç= é. Now Erdíos itroduces what is ow called the Deletio Method. This rado graph G alost surely has oly oèè edges i sall cycles ad every u vertices have at least edges. Take a speciæc graph G with these properties. Delete all the edges i sall cycles givig a graph G,. The certaily G, has o sall cycles. As fewer tha edges have bee deleted every u vertices of G,, which had ore tha edges i G, still have a edge. Thus the idepedece uber æèg, è ç u. But çèg, è ç æèg, è ç u ç ç As ca be arbitrarily large oe ca ow ake çèg, è ç k, copletig the proof. The use of coutig arguets becae a typographical ightare. Erdíos cosidered all graphs with precisely edges where = b 1+æ c. He eeded that alost all of the had the property that every u vertices èu as aboveè had ore tha. The uber of graphs failig that for a give set of size 4

9 evets beig utually idepedet over itegers x, settig ç l x ç 1= p x = K x where K is a large absolute costat. èfor the æitely ay x for which this is greater tha oe siply place x S.è Now fèè becoes a rado variable. For each xéywith x + y = let I xy be the idicator rado variable for x; y S. The we ay express fèè = P I xy. Fro Liearity of Expectatio Eëfèèë = X EëI xy ë= X p x p y ç K 0 l by a straightforward calculatio. Lets write ç = çèè = Eëf èèë. deviatio result. Oe shows, say, that The key igrediet is ow a large Prëfèè é 1 çë ée,cç Prëfèè é çë ée,cç where c is a positive absolute costat, ot depedet o; K or ç. This akes ituitive sese: as f èè is the su of utually idepedet rare idicator rado variables it should be roughly a Poisso distributio ad such large deviatio bouds hold for the Poisso. Now pick K so large that K 0 is so large that cç é l. Call a failure if either fèè é ç or fèè éç=. Each has probability less tha, failure probability. By the Borel-Catelli Lea èas P, covergesè alost surely there are oly a æite uber of failures ad so alost surely this rado S has the desired properties. While the origial Erdíos proof was couched i diæeret, coutig, laguage the use of large deviatio bouds ca be clearly see ad, o this cout aloe, this paper arks a otable advace i the Probabilistic Method : High Girth, High Chroatic Nuber Tutte was the ærst to show the existece of graphs with arbitrarily high chroatic uber ad o triagles, this was exteded by Kelly to arbitrarily high chroatic uber ad o cycles of sizes three, four or æve. A atural questio occured í could graphs be foud with arbitrarily high chroatic 3

10 Erdíos used a coutig arguetabove, i the ore oder laguage we would speak of the rado graph G ç Gè; pè with p = 1. The probability that G cotais a coplete graph of order k is less tha è! N,kèk,1è= é N k k k!,kèk,1è= é 1 ècalculatios as i the origial paperè ad so the probability that G or G 0 cotais a coplete graph is less tha oe so that with positive probability G does't have this property ad therefore there exists a G as desired. Erdíos has related that after lecturig o his result the probabilist J. Doob rearked ëwell, thats very ice but it really is a coutig arguet." For this result the proofs are early idetical, the probabilistic proof havig the ior advatage of avoidig the aoyig N èn,1è= factors. Erdíos writes iterchagably i the two styles. As the ethodology has progressed the probabilistic ideas have becoe ore subtle ad today it is quite rare to see a paper writte i the coutig style. We'll take the liberty of traslatig Erdíos's later results ito the ore oder style. The gap betwee k= ad 4 k for Rèk; kè reais oe of the ost vexig probles i Rasey Theory ad i the Probabilistic Method. All iproveets sice this 1947 paper have bee oly to saller order ters so that eve today li Rèk; kè 1=k could be aywhere fro p to 4, iclusive. Eve the existece of the liit has ot bee show! 1955: Sido Cojecture Let S be a set of positive itegers. Deæe fèè =f S èè as the uber of represetatios = x + y where x; y are distict eleets of S. We call S a basis if fèè é 0 for all suæcietly large. Sido, i the early 1930s, asked if there existed ëthi" bases, i particular he asked if for all positive æ there existed a basis with fèè =Oè æ è. Erdíos heard of this proble at that tie ad relates that he told Sido that he thought he could get a solutio i ëa few days". It took soewhat loger. I 1941 Erdíos ad Turça ade the stroger cojecture that there exists a basis with f èè bouded fro above by a absolute costat í a cojecture that reais ope today. I 1955 Erdíos ë?ë resolved the Sido cojecture with the followig stroger result. Theore: There exists S with fèè = æèl è. The proof is probabilistic. Deæe a rado set by Prëx Së =p x, the

11 THE ERD í OS EXISTENCE ARGUMENT Joel Specer The Probabilistic Method is ow a stadard tool i the cobiatorial toolbox but such was ot always the case. The developet of this ethodology was for ay years early etirely due to oe a: Paul Erdíos. Here we reexaie soe of his critical early papers. We begi, as all with kowledge of the æeld would expect, with the 1947 paper ë?ë givig a lower boud o the Rasey fuctio Rèk; kè. There is the a curious gap ècertaily ot reæected i Erdíos's overall atheatical publicatiosè ad our reaiig papers all were published i a sigle te year spa fro 1955 to : Rasey Rèk; kè Let us repeat the key paragraph early verbati. Erdíos deæes Rèk; lèas the least iteger so that give ay graph G of ç Rèk; lèvertices the either G cotais a coplete graph of order k or the copleet G 0 cotais a coplete graph of order l. Theore. Let k ç 3 The k= érèk; kè ç è! k, é 4 k,1 k, 1 Proof. The secod iequality was proved by Szekeres thus we oly cosider the ærst oe. Let N ç =. Clearly the uber of graphs of N vertices equals N èn,1è=.èwe cosider the vertices of the graph as distiguishable.è The uber of diæeret graphs cotaiig a coplete graph of order k is less tha è! N èn,1è= k é N k N èn,1è= kèk,1è= k! é N èn,1è= kèk,1è= sice by a siple calculatio for N ç k= ad k ç 3 N k ék! kèk,1è= But it follows iediately fro è*è that there exists a graph such that either it or its copleetary graph cotais a coplete subgraph of order k, which copletes the proof of the Theore. è*è 1