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

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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

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

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

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