The size Ramsey number of a directed path

Transcription

1 The size Ramsey umber of a directed path Ido Be-Eliezer Michael Krivelevich Bey Sudakov May 25, 2010 Abstract Give a graph H, the size Ramsey umber r e (H, q) is the miimal umber m for which there is a graph G with m edges such that every q-colorig of G cotais a moochromatic copy of H. We study the size Ramsey umber of the directed path of legth i orieted graphs, where o atiparallel edges are allowed. We give early tight bouds for every fixed umber of colors, showig that for every q 1 there are costats c 1 = c 1 (q), c 2 such that c 1 (q) 2q (log ) 1/q (log log ) (q+2)/q r e( P, q + 1) c 2 2q (log ) 2. Our results show that the path size Ramsey umber i orieted graphs is asymptotically larger tha the path size Ramsey umber i geeral directed graphs. Moreover, the size Ramsey umber of a directed path is polyomially depedet i the umber of colors, as opposed to the udirected case. Our approach also gives tight bouds o r e ( P, q) for geeral directed graphs with q 3, extedig previous results. 1 Itroductio Give a iteger q > 0, we write G (H) q if every q-colorig of E(G) cotais a moochromatic copy of H. The study of size Ramsey umbers (iitiated i [8]) is cocered with the followig questios. Give a graph H, what is the miimum umber of edges m for which there is a graph G with m edges such that G (H) q? Deote by r e (H, q) the size Ramsey umber of H with respect to colorig with q colors. That is, r e (H, q) = mi{ E(G) G (H) q }. The study of r e (K, q) is essetially equivalet to the study of the origial Ramsey umber. Namely, it ca be verified that if r e (K, q) = m the a clique with exactly m edges has the desired property. This result is attributed to Chvátal i [8]. School of Computer Sciece, Raymod ad Beverly Sackler Faculy of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel, idobee@post.tau.ac.il. Research supported i part by a ERC advaced grat. School of Mathematical Scieces, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel, krivelev@post.tau.ac.il. Research supported i part by USA-Israel BSF grat , by grat 1063/08 from the Israel Sciece Foudatio, ad by a Pazy memorial award. Departmet of Mathematics, UCLA, Los Ageles, CA bsudakov@math.ucla.edu. Research supported i part by NSF CAREER award DMS ad by a USA-Israeli BSF grat. 1

2 Beck proved [6] that for every costat q > 0, the size Ramsey umber of the udirected path o vertices is liear i, aswerig a questio of Erdős. Namely, there is a costat c q such that r e (P, q) c q. Later Alo ad Chug [3] provided a explicit costructio of graphs with this property. The liearity of the size Ramsey umber for bouded degree trees was proved by Friedma ad Pippeger [10]. Directed graphs. I this work we focus o the size Ramsey umber i directed graphs, ad i particular o the size Ramsey umber of a directed path o vertices. Rayaud [14] proved that every red-blue colorig of the complete symmetric directed graph (a directed graph i which betwee every two vertices there are edges i both directios) has a Hamilto cycle which is the uio of two moochromatic paths (a simple proof ca be foud i [12]). I particular, this shows that the size Ramsey umber of the path P for directed graphs with atiparallel edges is O( 2 ). Reimer [15] proved that every digraph with the property that every red-blue colorig has a path of legth must have Ω( 2 ) edges, therefore provig that Rayaud s boud for o-simple directed graphs is tight up to a costat factor. I this work we provide early tight bouds for the size Ramsey umber of directed paths i orieted graphs, where o parallel edges are allowed. We show that this umber is asymptotically larger tha the path size Ramsey umber whe such edges are allowed. Our approach also geeralizes to the case of o-simple directed graphs, ad we provide early tight bouds for such graphs for every costat q 3. Our first result is a lower boud o the size Ramsey umber. Theorem 1. For every q 1 there is a costat c 1 = c 1 (q) such that r e ( P, q + 1) c 1 2q (log ) 1/q (log log ) (q+2)/q. We also have the followig almost matchig upper boud. Theorem 2. There is a absolute costat c 2 such that for every q 1 r e ( P, q + 1) c 2 2q (log ) 2. Moreover, a radom touramet T N o N = Θ( q log ) vertices satisfies T N ( P ) q+1 with high probability. We stress that Theorem 1 proves that the size Ramsey umber of a directed path i orieted graphs is asymptotically larger tha the size Ramsey umber i geeral directed graphs for ay fixed umber of colors. I the course of the proof of Theorem 2, we actually prove the followig asymmetric Ramsey property. Theorem 3. There is a absolute costat c such that the followig holds. There is a orieted graph G with c 2 (log ) 2 edges such that every red-blue colorig of E(G) cotais a red path of legth or a blue path of legth log. 2

3 Our approach also gives matchig lower ad upper bouds for the case of directed o-simple graphs with more tha 2 colors, where atiparallel edges are allowed. This geeralizes the results of Rayaud [14] ad Reimer [15] to ay fixed q. Propositio 4. The size Ramsey umber of a directed path o vertices for q+1 colors i directed, o-simple graphs is Ω( 2q ). Propositio 5. The size Ramsey umber of a directed path o vertices for q+1 colors i directed, o-simple graphs is O( 2q ). Note that here we show that the size Ramsey umber (both for simple or o-simple directed graphs) of a directed path is polyomially depedet i the umber of colors, i cotrast to the udirected case where chagig the umber of colors chages the size Ramsey umber by a costat factor, as follows from the above metioed result of Beck [6]. Orgaizatio of the paper. The rest of the paper is orgaized as follows. I Sectio 2 we provide some otatio ad defiitios ad preset well kow theorems that will be used later o. I Sectio 3 we preset the proofs of our lower bouds, provig Theorem 1 ad Propositio 4. I Sectio 3 we prove the upper bouds, showig Theorem 2, Theorem 3 ad Propositio 5. Fially, i Sectio 5 we preset some cocludig remarks ad ope problems. Throughout the paper we assume that the uderlyig parameter is large eough. We do ot try to optimize costats, ad all logarithms are i base 2. We igore all floor ad ceilig sigs wheever these are ot crucial. 2 Prelimiaries All the graphs we cosider are directed. A orieted (or simple) graph is a graph with o atiparallel (or opposite) edges. Give a set of vertices A V, we deote by E(A) the set of edges i the iduced subgraph G[A]. The i-degree of a vertex v, deoted by d (v), is the umber of edges that are directed ito v, ad the out-degree of v, deoted by d + (v) is the umber of edges directed from v. The degree of v is d + (v) + d (v), ad we let (G) be the maximum degree i a graph G ad δ(g) be the miimum degree i G. For a vertex v we let N + (v) = {u V : (v, u) E} ad call this set the out-eighbors of v. We also let N (v) = {u V : (u, v) E} ad call this set the i-eighbors of v. For a set of vertices A we let N + (A) be a A N + (a). A set of vertices is acyclic if it does ot spa a cycle. The legth of a directed path is the umber of edges it cotais. The edge desity of a directed graph G = (V, E) is E. V 2 The complete graph o vertices, deoted by K, is a udirected graph for which every pair of vertices are coected. A complete symmetric directed graph is a o-simple directed graph where betwee every two vertices there are edges i both directios. A touramet is a orieted graph where betwee every two vertices there is a edge i exactly oe of the directios. A k-colorig of a graph is a mappig of the vertices to {1,..., k} such that every two adjacet vertices are mapped to distict values. The chromatic umber of a graph G, deoted by χ(g), is the miimum k such that G is k-colorable. It is well kow that every graph of maximum degree d is (d + 1)-colorable. A Hamilto cycle i G is a cycle that visits every vertex i G exactly oce. We will use the followig two theorems several times throughout the paper. The first oe, also metioed i the itroductio, is due to Rayaud [14] (see, e.g., [12] for a proof). 3

4 Theorem 2.1. Let G be a complete symmetric directed graph o t vertices. The every red-blue colorig of E(G) cotais a Hamilto cycle which is the uio of two moochromatic paths. I particular, it cotais a moochromatic path of legth t/2. We also eed the followig simple theorem that was proved idepedetly by Gallai [11] ad Roy [16] (see, e.g., [13], Chapter 9, Problem 9, for a proof). Theorem 2.2. Let G be a directed graph with o path loger tha t. The G is (t + 1)-colorable. 3 Lower bouds I this sectio we show that every orieted graph with relatively few edges has a edge colorig without a log moochromatic path. We first prove that every sparse orieted graph has a large acyclic set, ad the use it to show that every graph admits a partitio ito a relatively small umber of idepedet sets ad acyclic sets. We coclude by showig that give such a partitio we ca color the edges with o log moochromatic path. 3.1 Sparse graphs have large acyclic sets The followig lemma is well kow ad easy. Lemma 3.1. Every touramet o vertices has a acyclic set of size log + 1. Proof. We may assume that = 2 k for some iteger k, as otherwise we ca take ay subotouramet of size 2 log. We prove it by iductio o k, ad ote that the case k = 0 is trivial. Suppose that the iductio hypothesis is true for k, ad we ext prove it for k+1. Ideed, every touramet o 2 k+1 vertices cotais a vertex v with out-degree at least 2 k. By iductio hypothesis, N (v) cotais a acyclic set A of size k+1, ad thus A {v} is a acyclic set of size k+2, as required. Sice every orieted graph is a subgraph of a touramet, we get the followig direct cosequece. Corollary 3.2. Every orieted graph o vertices has a acyclic set of size log. Our mai result i this subsectio is that every sparse orieted graph has a larger acyclic set. This is a directed versio of a lemma by Erdős ad Szemerédi [9]. Lemma 3.3. There is a absolute costat c > 0 such that the followig holds. Every orieted graph G with vertices ad at most ε 2 edges cotais a acyclic set of size c log ε log (1/ε). Proof. Let G be a orieted graph with vertices ad ε 2 edges. We ca assume that ε < 1/4 as otherwise the lemma follows easily by takig c small eough ad applyig Corollary 3.2. Let G = (V, E) be the subgraph of G obtaied by removig every vertex with i-degree greater tha 2ε. Observe that we removed at most /2 vertices from G ad therefore V /2. Let U be a maximum acyclic set i G, ad assume that U < has i-degree at most 2ε, we have d (v) < 2ε U. v U 4 log 10ε log (1/ε). Sice every vertex

5 Let R be the set of vertices i V \ U with at least 5ε U out-eighbors i U, the R 2 5, as otherwise the total umber of edges directed ito U is more tha 2ε U. Let R = V \ (U R ) be the set of vertices outside U with at most 5ε U out-eighbors i U, ad we coclude that R /2 2/5 U /20. For every vertex v R we defie a set of vertices S v U of size exactly 5ε U that cotais N + (v) U. Usig the iequality ( ) k ( e k )k, we get that the total possible umber of subsets of U of this size is ( ) U ( e ) 5ε U log 2 2 log (1/ε) log e 5ε 1/2. 5ε U 5ε Therefore, by the pigeohole priciple, there is a set R R of size at least = 1/2 20 1/2 20 such that N + (R ) U 5ε U. Also, by Corollary 3.2, R cotais a acyclic set R R of size at least 1 2 (log 10). Note that R (U \ N + (R )) is a acyclic set of size at least R + U 5ε U U ( log 10 ) log 2 log (1/ε) U, assumig that ε < 1/4 ad is large eough. Therefore, we get a cotradictio as U is ot a maximum acyclic set, ad the lemma follows. 3.2 Acyclic colorigs ad colorig acyclic sets I this subsectio we give two buildig blocks that will be used later i the proof of Theorem 1. We first cosider the case where we are give a colorig of the edges i some iduced disjoit sets without a log moochromatic path, ad we wish to color the edges betwee them while keepig the legth of a logest moochromatic path relatively small. We the cosider the case whe we are give a acyclic set, ad show how to color its edges with o log moochromatic path. Give a set of vertices A ad a colorig ϕ of E(A), let l ϕ (A) be the legth of a logest moochromatic path of i A with respect to ϕ. We prove the followig. Lemma 3.4. Let A 1,..., A k be disjoit sets, ad let ϕ be a colorig of k i=1 E(A i) with (q + 1) colors such that for every 1 i k we have l ϕ (A i ) r. The ϕ ca be exteded to a (q + 1) colorig ϕ of E( k i=1 A i) such that ( k ) l ϕ A i q(r + 1) q k. i=1 Proof. Let s be the miimal iteger such that k s q. For each 1 i k, we deote by (i) the represetatio of i i base-s with exactly q digits. That is, we represet each idex as a vector of legth q, lettig (i) y be the y th coordiate of i, ad for each 1 y q, 0 (i) y s 1. We ow defie a acyclic colorig of the edges betwee the sets. For two sets A i ad A j, we color all the edges from A i to A j as follows. If there is a idex y such that (i) y < (j) y, we color all the edges i color y (if more tha oe choice of y is possible, choose oe arbitrarily). Otherwise, we color all the edges from A i to A j i color q

6 Note that by the defiitio of the ew colorig, if a moochromatic path leaves a set A i it will ever retur to this set. Also, if P is a moochromatic path colored by 1 y q, the the y th coordiates of all the sets visited by the path are all distict, ad therefore there are at most s sets i the path. For a moochromatic path P that is colored by (q + 1), if A j is visited by P after it visits A i the (i) y (j) y for every coordiate 1 y q, ad there is a strict iequality i at least oe coordiate. Therefore, the sums of the coordiates of all the sets visited by the path are all distict, ad hece such a path visits at most sq distict sets. We coclude that every moochromatic path colored by 1 y q visits at most s sq distict sets, ad every moochromatic path colored by q + 1 visits at most sq sets. A logest moochromatic path cotais at most r edges from each A i, plus oe edge that leaves A i. It visits at most qs = q q k sets. Therefore, the legth of this path is bouded by q(r + 1) q k, as claimed. I particular, we get the followig immediate corollary, that geeralizes a argumet of Reimer [15]. Corollary 3.5. Let G be a k-colorable graph. There is a (q+1)-colorig of E(G) with o moochromatic path loger tha q q k. We ext prove similarly that oe ca color the edges of a acyclic graph Z usig q colors with o moochromatic path loger tha q t, where t is the legth of a logest directed path i Z. Lemma 3.6. Let Z be a acyclic graph i which a logest directed path has t edges. There is a q-colorig of E(Z) with o moochromatic path loger tha q t + 1. Proof. Let s be the miimal umber such that t+1 s q, ad we prove that there a q-colorig with o moochromatic path loger tha s. For a vertex v let l Z (v) be the legth of a logest path i Z that eds i v, ad for each 0 i t let Z i = {v : l Z (v) = i}. Note that these sets are well defied sice Z is a acyclic graph. Moreover, each Z i is a idepedet set ad all the edges i the graph are directed from Z i to Z j for j > i. Let (i) be the ecodig of the umber i i base s with exactly q digits. Observe that for each j > i there is a idex 1 y q for which (j) y > (i) y. We ow defie the colorig as follows. For j > i, we color all the edges from Z i to Z j i color y where y is a idex such that (j) y > (i) y. If there is more tha oe feasible choice of y, we choose oe of them arbitrarily. Agai, we get that i every moochromatic path of color y, all the sets Z i that are visited by the path have distict y th coordiate, ad there is at most oe vertex from each Z i. We coclude that every moochromatic path cotais at most s vertices, as required. We will also eed the followig well kow ad simple claim, ad we give its proof for completeess. Claim 3.7. Let G be a udirected graph with m edges, tha G is 2 m-colorable. Proof. Every optimal vertex colorig cotais a vertex betwee every two color classes, hece m ( ) χ(g) 2 ad we get that χ(g) 2 m as required. 6

7 Proof of Theorem 1. Let G = (V, E) be a directed graph with c 1 2q (log ) 1/q (log log ) (q+2)/q edges. Defie X = {v V : d + (v) + d (v) ( ) q }. 2q Note that (G[X]) ( 2q )q (whe G[X] is cosidered here as a udirected graph), ad therefore, χ(g[x]) ( 2q )q + 1. By Corollary 3.5, there is a (q + 1)-colorig of E(X) with o moochromatic path loger tha /2 + q. Let Y = V \ X, ad Y = m. Sice E(G) c 1 2q (log ) 1/q ad every vertex i Y has at least (log log ) (q+2)/q ( 2q )q icidet edges, we have by double coutig m 2 E ( 2q )q 2c 1(2q) q q (log ) 1/q (log log ) (q+2)/q. (3.1) Cosider the followig procedure that partitios most of the vertices i Y ito Θ(log log ) families Y (1), Y (2),..., where each family is composed of ot too may acyclic sets. Roughly speakig, we partitio our vertices ito acyclic sets as follows. We maitai a idex i, ad at the i th step we fid acyclic sets that cover approximately m/2 i vertices, ad group these sets to a family Y (i). To this ed, at the begiig of each step we fix a umber a i for which we are guarateed by Lemma 3.3 that throughout the i th step we ca always fid a acyclic set of size a i. By the ed of the procedure, most of the vertices are partitioed ito families, ad each family is composed of acyclic sets of the same size. We iitiate the procedure by takig i, j = 1 ad Y = Y, where i represets the step umber. At the begiig of step i, we fix ε i = E(Y ) ad a (m/2 i ) 2 i = c log (m/2i ) ε i log (1/ε i ) (where c is a absolute costat give by Lemma 3.3), ad also set j := 1. The step eds whe Y m/2 i. At the i th step, we repeatedly fid a acyclic set Z ij i Y of size a i, ad set Y := Y \ Z i,j ad the icrease j by oe. The procedure termiates whe E(Y ) 2q. (16q) 2q We first stress that for every i, j we ca fid a acyclic set Z ij as claimed. Observe that durig each step, the value ε i is fixed, the umber of edges does ot icrease, ad the umber of vertices is at least m/2 i. Hece, ε i is a upper boud o edge desity durig the step. Therefore, Lemma 3.3 guaratees the existece of a acyclic set Z ij of size exactly a i. Let Y (i) = {Z ij } be the family of acyclic sets that is costructed at step i, ad let k i = Y (i). We ext boud the size of a logest moochromatic path i k i j=1 Z i,j for every i, ad start with i = 1. Let ε be the edge desity i Y. The ε = ε 1 /4. It is easy to verify that (sice clearly m 2 1/ε) ad hece the size of each acyclic set is at least c log (m/2) c(log m 1) = ε i log (1/ε i ) 4ε(log (1/ε) 2) c log m 4ε log (1/ε). log m 1 log (1/ε) 2 log m log (1/ε) As we complete the first step just after we cover at least m/2 vertices with acyclic sets, we get that the umber of acyclic sets satisfies k Moreover, sice all acyclic sets are of (m/2) 4ε log (1/ε) c log m the same size, the size of each Z 1j is bouded by m/2 k

8 By Lemma 3.6, we get that the edges of each acyclic set E(Z 1,j ), for every 1 j k 1, ca be colored with (q + 1) colors with o moochromatic path loger tha q+1 m/2 k We the ca apply Lemma 3.4 to color the edges betwee Z 1,j1 ad Z 1,j2 for each j 1, j 2. The umber of sets is k 1, ad the size of a logest moochromatic path i each of them is bouded by q+1 m/2 k We coclude that the edges of Z 1,j admit a (q + 1)-colorig with o moochromatic path loger tha q( q ( k 1 + 1) m/2 q+1 k m q ) 1 k 1 q(q+1) 2q. 2 q Note that the edge desity ε i Y satisfies ε E m 2 = c 1 2q (log ) 1/q m 2. (3.2) (log log ) (q+2)/q 2q (16q) 2q m2 E(Y ) m2 (16q) 2q Moreover, sice we assume that the umber edges is at least termiates before this step begis), we get that 1 ε = log (1/ε) log ( (16q) 2q assumig that is large eough. Therefore, we have the followig boud. 2q 4c2 1 (2q)2q 2q (log ) 2/q (log log ) (2q+4)/q m q k 1 2 q mq+1 ε log (1/ε) 2 q 2 c log m mq 1 c 1 2q (log ) 1/q log log 2 q 4 c log m (log log ) (q+2)/q 8(2q)q(q 1) c q 1 q(q+1) log log log (log log ) q+2 c log m 8(2q)q(q 1) c q 1 q(q+1) c(log log ) q+1 (as otherwise the procedure ad therefore by (3.1) 2q ) 2 log log, (3.3) Where the first iequality follows from the boud o k 1, the secod oe from substitutig ε ad 1/ε ad usig (3.2) ad (3.3), ad the third oe oe from substitutig m (ad applyig (3.1)). The last iequality follows from the boud log log m that clearly holds. c Takig c 1 < 1/q 8(2q) q (16q (we do ot try to optimize the depedece i q here), we get that 2 ) q+1 2q ( m q ) 1 k 1 q(q+1) 1 2 q 8q q log log. It is ot difficult to verify that takig smaller values of m oly decreases the last expressio. Hece, by repeatig the same colorig method, we get that for each i, there is a (q + 1)-colorig of the edges spaed by the uio of the sets i Y (i) with o moochromatic path loger tha 1 8q q log log. Note that the umber of vertices i Y is decreased by a factor of at least 2 i each 8

9 step. Moreover, the procedure termiates whe E(Y ) 2q, ad thus it essetially termiates (16q) 2q if Y q (16q). Hece by applyig (3.1) we get that the umber of families is bouded by q log (16q)q m q log ( (16q) q q 2c 1(2q) q q (log ) 1/q (log log ) (q+2)/q ) log log, assumig agai that is large eough. Let W = i,j Z ij the set of vertices that were covered throughout the procedure. We color all the edges i E(W) whose edpoits belog to sets from distict Y (i) s with (q + 1) colors accordig to Lemma 3.4. Sice the legth of each moochromatic path spaed by the sets of each family is bouded by 1 8q q log log, ad the umber of families is bouded by log log, we get that the iduced subgraph of all vertices that are cotaied i sets from Y (i) admits a (q + 1)-colorig with o path loger tha ( ) 1 q 8q q + 1 ( q log log + 1 ) /4. log log Fially, after the procedure termiates, we are left with a set Y that satisfies E(Y ) By Claim 3.7, we get that G[Y 2 ] is q (16q) -colorable. Hece Corollary 3.5 implies that E(Y ) admits q 2q. (16q) 2q a (q + 1)-colorig with o moochromatic path loger tha / We therefore foud a partitio of V (G) ito three sets X, Y, W. The set E(X) admits a (q +1)- colorig with o moochromatic path loger tha /2 + q. The set E(Y ) admits a colorig with o moochromatic path loger tha /8, ad E(W) admits a colorig with o moochromatic path loger tha /4. We color all the edges that are either from X to Y or from X to W or from Y to W by the first color, ad all the edges that are either from Y to X or from W to X or from W to Y by the secod color. Every moochromatic path i E(G) that leaves oe of these three sets does ot retur to this set. Therefore, the legth of a logest moochromatic path is bouded by the sum of the legths of logest moochromatic paths i X, Y, W, plus at most two edges betwee them. Thus E(G) admits a colorig with o moochromatic path loger tha /2 + q + / /4 + 2 <, ad Theorem 1 follows. Proof of Propositio 4. Let G be a o-simple directed graph with ( ) 2q 3q edges. By Claim 3.7, G is 2 ( ) q-colorable, 3q ad hece by Corollary 3.5 it admits a (q+1)-colorig with o moochromatic path loger tha, ad the propositio follows. 4 Upper Bouds I this sectio we provide a orieted graph for which every q-colorig of its edges cotais a log moochromatic path. We start with the case of two colors. I Subsectio 4.1 we defie the otio of a k-pseudoradom digraph, ad show that a radom touramet o vertices is Θ(log )-pseudoradom with high probability. We ext show that every red-blue colorig of a k-pseudoradom digraph o vertices cotais a directed red path of legth Ω( k ) or a directed blue path of legth Ω(). This proves Theorem 3. We coclude this sectio by showig how to reduce the case of ay fixed umber of colors to the case of two colors, provig Theorem 2 ad 9

10 Propositio 5. For otatioal coveiece we use i this sectio to deote the umber of vertices i a Ramsey digraph G, rather tha the legth of a target path P as i Theorem 2, Theorem 3 ad Propositio Pseudoradom digraphs We start with the followig atural defiitio for k-pseudoradomess of directed graphs. Defiitio 4.1. We say that a directed graph G is k-pseudoradom if for every two disjoit sets A, B such that A, B k, there is at least oe edge of G from A to B. We first show the existece of a Θ(log )-pseudoradom digraph by showig that a radom touramet satisfies this property with high probability. Claim 4.2. A radom touramet o vertices is 2 log -pseudoradom with high probability. Proof. Let T be a radom touramet ad fix two disjoit sets A, B of size k i T. Sice every edge is orieted i each way uiformly ad idepedetly of the other edges, the probability that all the edges are directed from B to A is exactly 2 k2. There are at most ( ) 2 k 2k 22k log k log k (k!) 2 choices of pairs of sets of size k, ad thus by the uio boud the probability that there are two sets of size k for which all the edges are orieted i oe directio is bouded by 2 2k log k2 k log k = o(1), for k = 2 log. We also eed the followig property of k-pseudoradom digraphs. Claim 4.3. Let G be a k-pseudoradom directed graph, ad let A 1, A 2,..., A t be disjoit sets, each of size at least 2k. The there is a directed path v 1 v 2... v t, where for each 1 i t, v i A i. Proof. We say that a vertex u j A j is good if there is a directed path u j u j+1... u t such that u s A s, s = j,..., t. Clearly, our goal is to prove that there is a good vertex i A 1. Deote by A j the set of good vertices i A j. By defiitio, every vertex i A t is good, ad thus A t 2k. Also, if u j+1 A j+1 ad there is a edge from u j to u j+1, the u j A j. Usig a reverse iductio, assume that we kow that for some j t, A j k, the sice G is k-pseudoradom all but at most k of the vertices i A j 1 have a edge to A j. Therefore, all the vertices at most but k i A j 1 are actually i A j 1, ad thus A j 1 k. We coclude that A 1 k ad thus there is a path as required. The followig lemma shows that every k-pseudoradom graph cotais a log path. The proof follows ideas from [7, 5]. Lemma 4.4. Let G a k-pseudoradom orieted graph o vertices. The G cotais a directed path of legth 2k + 1. Proof. Recall that the DFS (Depth First Search) is a graph algorithm that visits all the vertices of a (directed or udirected) graph G as follows. It maitais three sets of vertices, lettig S be the set of vertices which we have completed explorig them, T be the set of uvisited vertices, ad U = V (G) \ (S T ), where the vertices of U are kept i a stack (a last i, first out data structure). 10

11 It is also assumed that some order σ o the vertices of G is fixed, ad the algorithm prioritizes vertices accordig to σ. The DFS starts with S = T = ad U = {σ 1 }. While there is a vertex i V (G) \ S, if U is o-empty, let v be the last vertex that was added to U. If v has a eighbor u T, the algorithm iserts u to U ad repeats this step. If v does ot have a eighbor i T the v is popped out from U ad is iserted to S. If U is empty, the algorithm chooses a arbitrary vertex from T ad pushes it to U. We are ow proceed to the proof of the lemma. We Execute the DFS algorithm. We let agai S, T, U be three sets of vertices as defied above. At the begiig of the algorithm, all the vertices are i T, ad at each step a sigle vertex either moves from T to U or from U to S. At the ed of the algorithm, all the vertices are i S. Therefore, at some poit we have S = T. Observe crucially that all the vertices i U form a directed path, ad that there are o edges from S to T. We coclude that S k 1, ad therefore U 2k + 2, so there is a directed path with 2k + 1 edges i U, as required. 4.2 The case of two colors I this subsectio we prove that every k-pseudoradom directed graph o vertices has a moochromatic red path of legth Ω( k ) or a moochromatic blue path of legth Ω(), ad this will prove Theorem 3. We prove the followig mai lemma. Lemma 4.5. Let G be a k-pseudoradom directed graph o vertices. The every red-blue colorig of its edges yields a red directed path of legth or a blue directed path of legth /28. 28k Proof. Fix a red-blue colorig of E(G). Let G R be the red graph, that cotais oly the red edges. If G R cotais a directed path of legth 14k, we are doe. Otherwise, by the Gallai-Roy theorem (Theorem 2.2), G R is 14k -colorable ad therefore has a partitio ito 14k idepedet sets. We partitio these idepedet sets ito sets of size exactly 7k, roudig dow the remaiig vertices (i particular, we remove every idepedet set smaller tha 7k). Sice we remove at most 7k 14k = /2 vertices, we remai with t 14k idepedet sets B 1, B 2,... B t, each cotaiig exactly 7k vertices. Note that each B i, 1 i t, spas a k-pseudoradom graph ad cotais oly blue edges. Therefore, by Lemma 4.4, each B i cotais a blue path of legth at least 5k. Sice there is a directed edge from the last k vertices of this path to the first k vertices i the path, we coclude that each B i cotais a directed blue cycle C i of legth at least 3k. We ext costruct a auxiliary graph H o t vertices, each vertex correspods to a cycle C i, ad with a slight abuse of otatio we deote these vertices by C 1, C 2,..., C t. The graph H is a complete symmetric directed graph, ad we color the edges from C i to C j by blue if there are at least k vertices i C i that have blue edges directed towards C j, ad red otherwise. Sice H is complete ad symmetric, by Rayaud s theorem (Theorem 2.1), it has a moochromatic path of legth t/2 28k. Let C i 1, C i2,..., C it/2 be the vertices i H alog the path, each such vertex represets a cycle. We complete the proof by cosiderig the followig two cases. H cotais a red path of legth t/2. For each C ij there is a set R ij for which oly red edges are goig towards C ij+1. Moreover, for every 1 j t/2 we have R ij 2k. Observe that all the edges from R ij to R ij+1 are red, ad therefore by Claim 4.3, there is a red path of legth t/2 28k with exactly oe vertex from each R ij, as required. 11

12 H cotais a blue path of legth t/2. Call a vertex i C ij a edpoit if it has a blue edge towards C ij+1. By the assumptio, there are at least k edpoits i C ij for every 1 j t/2, ad therefore from each vertex i C ij there is a path of legth at least k 1 that eds at some edpoit, i which we ca travel alog oe additioal edge towards C ij+1. We costruct a blue path of legth 28 by takig a arbitrary path of legth k 1 that eds at some edpoit i C i 1, movig through the edpoit to C i2, walkig through such a path to a edpoit of C i2 ad so o till we arrive at C it/2, where we ca agai walk alog a path of legth at least 3k 1 (that visits all the vertices i C it/2 ). We coclude that there is a blue path of legth at least k 28k = 28, as claimed. Lemma 4.5 ad Theorem 3 follow. Explicit costructios. Give a explicit costructio of a k-pseudoradom touramet o vertices, our approach shows that every red-blue colorig of such a touramet has a moochromatic path of legth Ω( k ). To the best of our kowledge, the best costructio of a k-pseudoradom touramet is give by Quadratic Residue touramets, defied as follows (see [4], Chapter 9). Let p 3 mod 4 be a prime. The vertices of the touramet T p are all the elemets i the fiite field Z p. For two vertices i ad j, there is a edge from i to j if ad oly if i j is a quadratic residue. It ca be show that sice p 3 mod 4, 1 is a quadratic oresidue ad therefore for each i, j there either there is a edge from i to j or a edge from j to i but ot both. This costructio gives k = Θ( ) (see [2] ad [4]). However, every red-blue colorig of ay touramet yields a moochromatic path of legth 1. To prove this, observe that if both the red ad blue graphs, formed respectively by takig all red ad blue edges i some fixed colorig do ot have a path of legth 1, the by the Gallai-Roy theorem (Theorem 2.2) they are both ( 1)-colorable. It is easy ad well kow (see, e.g., [13], Chapter 9, Problem 3) that give two graphs G 1, G 2 we have χ(g 1 G 2 ) χ(g 1 ) χ(g 2 ). We coclude that K, which is the uio of the red ad blue graph is ( 1)-colorable, ad we get a cotradictio. Hece, oly explicit costructios of o( )-pseudoradom graphs will be iterestig for our problem. 4.3 The geeral case Here we prove by iductio o q that for every k-pseudoradom directed graph G o 28k q vertices we have G ( P ) q+1. Theorem 2 will clearly follow by the fact that a radom touramet is Θ(log )-pseudoradom with high probability (Claim 4.2). The base case of the iductio (q = 1) follows directly from Lemma 4.5. Suppose that the result holds for q colors ad we ext prove it for q + 1 colors. Ideed, let G be a k-pseudoradom graph o 28k q vertices. Fix a colorig of E(G) with the colors 1, 2,..., q + 1. Deote by G q+1 G the subgraph with all edges that are colored q + 1. If G q+1 cotais a moochromatic path of legth we are doe. Otherwise, by the Gallai-Roy Theorem (Theorem 2.2), we get that G q+1 is -colorable ad hece cotais a idepedet set A of size 28kq = 28k q 1. Note that A spas a subgraph of G ad hece G[A] is k-pseudoradom. Also, the colors of all the edges of G[A] are amog {1, 2,..., q}. Therefore, by the iductio hypothesis G[A] ( P ) q, ad we coclude that G cotais a moochromatic path of legth, as desired. 12

13 Proof of Propositio 5. The proof for o-simple directed graphs follows similar lies. Note that the Gallai-Roy theorem is valid for o-simple directed graphs as well. Therefore, we ca use the same iductio o q. The base case (q = 1) follows from Rayaud s theorem (Theorem 2.1). Suppose that the result holds for q colors, the correctess for q + 1 colors follows by takig a complete symmetric graph o q vertices, ad cosiderig the subgraph with all edges colored by the (q + 1) th color. If this graph cotais a directed path of legth we are doe, otherwise we fid a iduced subgraph of order q 1 i which all edges are colored 1, 2,..., q, ad applyig the iductio hypothesis, Propositio 5 follows. 5 Cocludig remarks ad ope problems We proved early tight bouds for the size Ramsey umber of a directed path for orieted graphs. We proved that every red-blue colorig of the edges of a k-pseudoradom graph o vertices cotais a red path of legth Ω( k ) or a blue path of legth Ω(), but it might be the case that this approach ca also give better symmetric Ramsey bouds. A iterestig questio is whether every red-blue colorig of a k-pseudoradom graph cotais a moochromatic path of legth Ω( k ). Clearly every progress i this directio will improve our upper bouds. Aother related questio is about the asymptotic behavior of the maximum legth of a moochromatic path i every red-blue colorig of a radom touramet. Here we proved that every touramet T has a colorig with o moochromatic path loger tha O( log ), ad also that with high probability a radom touramet T has a moochromatic path of legth Ω( log ) i every red-blue colorig. It would be very iterestig to close the gap betwee these bouds. Whe provig the lower boud o the size Ramsey umber, we study the miimal umber k for which a certai graph ca be partitioed ito k acyclic sets. This parameter was studied, e.g., i [1], ad it was cojectured that i every orieted graph G = (V, E) there is a acyclic set of size V 2 (1 + o(1)) E log E V. If this cojecture is true, the our lower boud ca be slightly improved. It is easy to verify that there is o k-pseudoradom orieted graph o vertices for k log 2, as every such graph has a acyclic set of size log ad therefore has two sets of size log 2 with o edges i oe of the directios. O the other had we proved that for k = 2 log such graphs do exist. Hece, it will be iterestig to determie the miimum k for which there is a k-pseudoradom orieted graph o vertices. Aother appealig questio is to provide better explicit costructios of k-pseudoradom orieted graphs. Refereces [1] R. Aharoi, E. Berger ad O. Kfir, Acyclic systems of represetatives ad acyclic colorigs of digraphs, J. of Graph Theory 59 (2008), [2] N. Alo, Eigevalues, geometric expaders, sortig i rouds ad Ramsey theory, Combiatorica 6 (1986), [3] N. Alo ad F. R. K. Chug, Explicit costructio of liear sized tolerat etworks, Discrete Math. 72 (1988), [4] N. Alo ad J. Specer, The probabilistic method, Third editio, Wiley,

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