1. Introduction. All graphs considered here are finite and simple. For a graph G =(V,E), let Δ(G),δ(G), and d(g) = 2 E

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1 SIAM J. DISCRETE MATH. Vol., No. 4, pp c 008 Society for Idustrial ad Applied Mathematics LARGE NEARLY REGULAR INDUCED SUBGRAPHS NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV Abstract. For a real c 1 ad a iteger, let f(, c) deote the maximum iteger f such that every graph o vertices cotais a iduced subgraph o at least f vertices i which the maximum degree is at most c times the miimum degree. Thus, i particular, every graph o vertices cotais a regular iduced subgraph o at least f(, 1) vertices. The problem of estimatig f(, 1) was posed log ago by Erdős, Fajtlowicz, ad Stato. I this paper we obtai the followig upper ad lower bouds for the asymptotic behavior of f(, c): (i) For fixed c>.1, 1 O(1/c) f(, c) O(c/ log ). (ii) For fixed c =1+ε with ε>0 sufficietly small, f(, c) Ω(ε / l(1/ε)). (iii) Ω(l ) f(, 1) O( 1/ l 3/4 ). A aalogous problem for ot ecessarily iduced subgraphs is briefly cosidered as well. Key words. iduced subgraphs, regular subgraphs, Ramsey-type problem AMS subject classificatio. 05C35 DOI / Itroductio. All graphs cosidered here are fiite ad simple. For a graph G =(V,E), let Δ(G),δ(G), ad d(g) = E V deote its maximum degree, miimum degree, ad average degree, respectively. The desity of G is p = E / ( ) V ; clearly, this is a umber betwee 0 ad 1. For U V, let G[U] deote the subgraph of G iduced o U. Defiitio 1. A graph G is c-early regular if Δ(G) c δ(g). For a graph G =(V,E) ad a costat c 1, let Defie f(g, c) = max{ U : G[U] isac-early regular graph}. f(, c) = mi{f(g, c) : V (G) = }. Thus, every graph G o vertices cotais a c-early regular iduced subgraph o at least f(, c) vertices. I particular, for c = 1 every such G cotais a strictly regular iduced subgraph o at least f(, 1) vertices. The problem of estimatig f(, 1) was posed by Erdős, Fajtlowicz, ad Stato (cf. [3] or [, page 85]). By the kow estimates for graph Ramsey umbers (cf., Received by the editors October 9, 007; accepted for publicatio (i revised form) April 8, 008; published electroically September 4, Schools of Mathematics ad Computer Sciece, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel, ad Istitute for Advaced Study, Priceto, NJ (ogaa@tau.ac.il). The research of this author was supported i part by the Israel Sciece Foudatio, by a USA-Israeli BSF grat, by the Herma Mikowski Mierva Ceter for Geometry at Tel Aviv Uiversity, ad by the Vo Neuma Fud. Departmet of Mathematics, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel (krivelev@post.tau.ac.il). The research of this author was supported i part by USA-Israel BSF grat 0063 ad by grat 56/05 from the Israel Sciece Foudatio. Departmet of Mathematics, UCLA, Los Ageles, CA 90095, ad Istitute for Advaced Study, Priceto, NJ (bsudakov@math.ucla.edu). The research of this author was supported i part by NSF CAREER award DMS , NSF grats DMS ad DMS , by a USA- Israeli BSF grat, ad by the State of New Jersey. 135

2 136 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV e.g., [6]), every graph o vertices cotais either a clique or a idepedet set of order Ω(l ). This implies that f(, 1) Ω(l ). Erdős, Fajtlowicz, ad Stato cojectured that the ratio f(, 1)/ l teds to ifiity as teds to ifiity. We are uable to prove or disprove this cojecture ad ca oly obtai several bouds, which are listed i the followig results. The first deals with the case of large c. Propositio 1.1. There exists a absolute costat b so that for all K.1, f(, K) 1 b/k. The problem of obtaiig a otrivial lower boud for values of c close to 1 is more iterestig. Here we first deal with the case of graphs with positive desity ad show that ay such graph must cotai a early regular subgraph o a liear umber of vertices. Theorem 1.. Let ε>0 be a small real, ad let p satisfy 0 <p<1. The, for every sufficietly large, ay graph G =(V,E) o vertices with desity at least p cotais a iduced (1 + ε)-early regular subgraph o at least ( ε ε 0.5 6)144 l( 1 p ) vertices. For geeral (possibly sparse) graphs we have the followig. Theorem 1.3. Let ε>0 be a sufficietly small costat. The f(, 1+ε) ε 50 l(1/ε) for all sufficietly large. Our upper bouds for f(, c) are rather far from the lower bouds. For the strictly regular case we prove the followig. Theorem 1.4. f(, 1) O( 1/ log 3/4 ). This is a slight improvemet of a earlier estimate of Bollobás (cf. []), who showed that for every ɛ>0, f(, 1) c(ɛ) 1/+ɛ. For the early regular case we have the followig. Propositio 1.5. For every costat K, f(, K) 7K log. The lower bouds are proved i the ext sectio, ad the upper bouds are preseted i sectio 3. We coclude i sectio 4 with a few ope problems ad a brief discussio of a aalogous problem for ot ecessarily iduced subgraphs. Throughout this paper we assume, wheever eeded, that the umber of vertices of the graphs discussed is sufficietly large. To simplify the presetatio, we make o attempt to optimize the absolute costats, ad we omit all floor ad ceilig sigs wheever these are ot crucial. We also use the followig stadard asymptotic otatio: For two fuctios f(), g() of a atural valued parameter, we write f() =o(g()) wheever lim f()/g() =0;f() =O(g()) if there exists a costat C > 0 such that f() Cg() for all ; ad f() =Ω(g()) if g() = O(f()).. Lower bouds. For a graph G =(V,E) ad a subset U V, the umber of edges of G spaed by U i G is deoted by e G (U); the umber of edges betwee disjoit subsets U, W of vertices of G is deoted by e G (U, W )..1. Large c. I this subsectio we prove Propositio 1.1, which provides a lower boud for f(, c) whe c is a relatively large costat. We eed the followig rather stadard argumet, allowig oe to pass from a graph with a large average degree to oe with a large miimum degree.

3 LARGE NEARLY REGULAR INDUCED SUBGRAPHS 137 Propositio.1. Let K > 1,α < 1/ be costats. The every graph G = (V,E) o V = vertices with Δ(G) Kd(G) cotais a iduced (K/α)-early regular subgraph G with at least 1 α K Kα K α vertices ad at least K 4α d edges. Proof. Deote the average degree d(g) ofg by d. We ca obviously assume that d>0. Start with G ad delete repeatedly vertices of degree less tha αd util there are oe left. Deote the resultig graph by G. The G is a iduced subgraph of G, satisfyig Δ(G ) Δ(G), δ(g ) αd, implyig Δ(G ) Δ(G) Kd K α δ(g ), ad thus G isa(k/α)-early regular graph. We ow estimate the umber of vertices of G. Deote the latter by t. While creatig G from G, we deleted less tha ( t)αd edges, ad thus Δ(G ) d(g ( E(G) ( t)αd) ) > t But Δ(G ) Δ(G) Kd, implyig = d ( t)αd t. d ( t)αd Kd. t Solvig the above iequality for t, we get t 1 α K α, supplyig the required lower boud for the umber of vertices of G. To boud the umber of its edges, ote that the umber of vertices deleted is t, ad hece the umber of edges deleted is at most ( t)αd, leavig at least ( ) ( ) d ( t)αd = α d + tαd α d + 1 α K Kα αd = K α K 4α d, as eeded. Remark 1. It is istructive to observe that the above argumet breaks dow completely for α 1/. Therefore, whe estimatig f(, c) from below for small c, i particular for c<, we will adapt a differet strategy. We proceed with the followig result, whose proof resembles that of oe of the results i [4]. Propositio.. Let K>1be a costat. Every graph G =(V,E) o V = V (G) = vertices cotais a iduced subgraph G o at least 1+log (1 1 K ) vertices, for which Δ(G ) Kd(G ). For large K the above estimate behaves like 1 Θ( 1 K ). Therefore, the assertios of Propositios.1 ad. imply that of Propositio 1.1. Proof of Propositio.. Set G 0 = G, k = log. For i =0,...,k repeat the followig loop. Set i = V (G i ), Δ i =Δ(G i ), d i = d(g i ). If Δ i Kd i, abort the loop. Otherwise, delete repeatedly vertices of degree at least Kd i / from G i util there are oe left. Let G i+1 be the resultig graph ad icremet i. Deote by G the resultig graph of the above described process. Observe that at iteratio i we delete at most E(G i ) /(Kd i /)=( i d i /)(Kd i /) = i /K vertices,

4 138 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV ad thus i+1 (1 1/K) i. It follows that ( V (G ) 1 1 ) k = 1+log (1 1 K ). K If G was created whe the above loop was aborted due to Δ(G i ) Kd(G i ), the obviously the obtaied graph satisfies the assertio of the theorem. Otherwise, G was obtaied after k iteratios. At each such iteratio i, we have Δ i+1 Kd i / ad d i Δ i /K, implyig Δ i+1 Δ i /. Therefore, i this case Δ(G ) Δ 0 ( ) k 1 < ( ) log 1 =1, implyig that G has o edges, ad thus Δ(G )=d(g )=0,adG ca agai serve as the required graph... Small c. Next we treat the more challegig case where the costat c i f(, c) is very close to 1. Throughout this subsectio ε deotes a small positive real. We start with several lemmas. Lemma.3. Let G =(V,E) be a graph o vertices with desity p, ad let ε>0. The G cotais a iduced subgraph G of desity p p o a set of ε ε l( 1 p ) vertices so that every set of t ε vertices of G spas at most ( t ) p (1 + ε) edges. Proof. Set G 0 = G. Fori =0, 1,... repeat the followig loop. Set i = V (G i ), m i = e(g i ), p i = m i ). If G i cotais a subset U V (G i ) of at least ε i vertices such that e Gi (U) ( U ) pi (1 + ε), the set G i+1 := G i [U], i := i +1. Observe that after k iteratios of the above loop, the desity p k of the curret graph G k satisfies p k (1+ε) k p 0. Thus, if the loop is repeated at least ε l( 1 p ) times, we have p k (1 + ε) ε l( 1 p ) p>1, which is a cotradictio. It follows that the above process cocludes after less tha ε l( 1 p ) iteratios. The resultig graph G k has k vertices ad m k edges. Observe that at each iteratio the umber of vertices of the ew graph is at least a ε-proportio of the umber of vertices of the previous graph. Therefore, k ε k V (G) ε ε l( 1 p ). We ca thus take G = G k, = k to complete the proof. Lemma.4. Let G =(V,E) be a graph o V = vertices with m = E edges ad desity p = m/ ( ) a for some costat 0 <a<1. Suppose that ( ) t (1) every t ε vertice i G spas at most p(1 + ε) edges. The for every subset U V of cardiality U = ε i G, there are at most ε p(1+ ε) ( i

5 LARGE NEARLY REGULAR INDUCED SUBGRAPHS 139 edges betwee U ad its complemet i G. Proof. Assume that U V cotradicts the above statemet. Deote e 1 = e G (U), e = e G (U, V U), e 3 = e G (V U). The e 1 + e + e 3 = m = ( ) p. Choose uiformly at radom a subset X V U of cardiality X = x = ε V U = ε(1 ε). The the expected umber of edges of G spaed by U X is E[e G (U X)] = e 1 + x V U e x(x 1) + V U ( V U 1) e 3 >e 1 + x ε e x ( + ( ε) 1 1 ) e 3 x = e 1 + εe + εe 3 O() =e 1 + εe + ε(m e 1 e ) O() ( ε ε)e + εm O() ( ε ε)ε p(1+ ε)+εm O() =( ε ε)ε(1+ ε) p + ε p O() =:A. O the other had, by the assumptio o G, every such set U X satisfies ( ) ε + ε(1 ε) e G (U X) p(1 + ε) ε p (1 + ε ε) (1 + ε) = ε p (1+ ε ε ε 3/ + ε )(1 + ε) = ε p (1+ ε + O(ε 5/ )) =: B. Let us compare the asymptotic (i small ε) behavior of the two quatities A ad B defied above. We have A = ε p (1+( ε ε)(1+ ε)) O() = ε p (1+ ε +ε 4ε 3/ ) O(). Sice = o( p), we have that A>Bfor ε small eough, which is a cotradictio. Lemma.5. Let G = (V,E) be a graph o V = vertices with m = E edges ad desity p = m/ ( ) a for some costat 0 <a<1. Suppose that (1) holds. The, for all sufficietly large, G cotais a iduced subgraph G o at least (1 ε ε) >/ vertices with maximum degree Δ(G ) (1+3 ε)p ad miimum degree δ(g ) (1 ε)p. I particular, G is c-early regular for c =(1+6 ε). Proof. Let U be a set of ε vertices of highest degrees i G (ties are broke arbitrarily). Set H = G[V U]. We claim that all vertex degrees i H are at most

6 1330 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV p(1+3 ε). If this is ot so, the the degrees of all vertices of U i G are at least p(1+3 ε), implyig (through coditio (1)) e G (U, V U) ε p(1+3 ε) e G (U) ε p(1+3 ε) ε p(1 + ε) >ε p(1+ ε) for small eough ε, thus cotradictig Lemma.4. Therefore, H is a iduced subgraph of G o V (H) =(1 ε) vertices, of maximum degree Δ(H) p(1+3 ε), still satisfyig coditio (1), ad havig () E(H) ( )p ε p(1 + ε) ε p(1+ ε) ( ) p ε p edges. We ow repeatedly delete from H vertices of degree less tha p(1 ε) util there are o such vertices, or util we have deleted ε of them. Assume the latter case ad deote the set of ε deleted vertices by W. The the set V (H) W has V (H) W =(1 ε ε) vertices ad by () spas at least e H (V (H) W ) E(H) ε p(1 ε) = ( ) p ε p ε p(1 ε) ( ) p ε p +ε p = p ( ) 1 4 ε +4ε O(p) edges. O the other had, by coditio (1), the set V (H) W satisfies ( ) (1 ε ε) e H (V (H) W ) p(1 + ε) (1 ε ε) (1 + ε) p = p ( 1 4 ) ε +3ε + O(ε 3/ ). Comparig the above two estimates for e H (V (H) W ), we get a cotradictio for small eough ε. It follows that the above deletio process stops before ε vertices have bee deleted. Deote the resultig graph by G. The G has V (G ) ε ε > vertices ad has maximum degree Δ(G ) p(1+3 ε) ad miimum degree δ(g ) p(1 ε). Hece Δ(G ) 1+3 ε 1 ε δ(g ) < (1+6 ε)δ(g ), completig the proof of the lemma. We are ow ready to prove Theorems 1. ad 1.3.

7 LARGE NEARLY REGULAR INDUCED SUBGRAPHS 1331 Proof of Theorem 1.. By Lemma.3 (with ε 36 a iduced subgraph G of desity p p o playig the role of ε), G cotais ( ) 7 ε ε l( 1 p ) ( ε ε = 36 6)144 l( 1 p ) vertices, such that every set of t ε 36 vertices of G spas at most ( ) t p (1 + ε 36 ) edges. Sice p p> 1/ (as p>0is a costat ad is large), Lemma.5 implies that G cotais a iduced subgraph G o at least / vertices which is c-early-regular for c = Proof of Theorem 1.3. Set ε 36 =1+ε, as eeded. ε 0 = ε 36, a = ε 0 3 l(1/ε 0 ). Let G =(V,E) be a graph o vertices. Deote by p = E / ( ) the desity of G. The average degree of G is at most p ad by Turá s theorem G cotais a idepedet set U of size /(p + 1). Therefore we ca assume that p a, as otherwise, p +1 1 a +1 (1 o(1))a =(1 o(1)) =(1 o(1)) ε 108 l(36/ε ε ) > 50 l(1/ε), ε 0 3l(1/ε 0 ) where here we used the assumptios that ε is sufficietly small ad is sufficietly large. This gives a iduced 0-regular subgraph of G, ad we ca thus ideed assume that p a. By Lemma.3, G cotais a iduced subgraph G of desity p p o ε l( a ) 0 3l(1/ε 0 ) l ε0 = ε0 = 1 3 vertices, i which the desity of the iduced subgraph o ay set of at least ε 0 vertices does ot exceed p (1 + ε 0 ). By Lemma.5, G (ad hece G) cotais a iduced subgraph o at least /3 vertices, which is (1 + 6 ε 0 )=(1+ε)- early regular, completig the proof. Remark. Note that we have actually proved the followig result, which is stroger tha the assertio of Theorem 1.3: Every graph o vertices cotais either ε a idepedet set of size at least 50 l(1/ε),ora(1+ε)-early regular iduced subgraph o at least 0.5 1/3 vertices. 3. Upper bouds The strictly regular case. Proof of Theorem 1.4. Fix a iteger k satisfyig k C 1/ l 3/4, where C>0 is a sufficietly large costat to be set later. We will work with the followig model of radom graphs o vertices which we deote by G(, p). Let p =(p 1,...,p ), where p i = i, i =1,...,.

8 133 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV The G(, p) is the probability space of graphs with vertex set [] = {1,..., }, where for each pair 1 i j, (i, j) is a edge of G(, p) with probability p i p j, idepedetly of all other pairs. Notice that the probability of each idividual pair (i, j) to be a edge of G(, p) is strictly betwee 1/16 ad 9/16. Propositio 3.1. Let X 1,...,X t be idepedet Beroulli radom variables, where Pr[X i =1]=ρ i, i =1,...,t. Let X = X X t. Assume that 1/16 ρ i 9/16 for i =1,...,t. The for every iteger 0 s t, Pr[X = s] c 0 / t for some absolute costat c 0 > 0. Proof. For every 1 i t, we represet X i as a product X i = Y i Z i, where {Y i,z i } is a collectio of mutually idepedet Beroulli radom variables defied by Pr[Y i =1]= 9 16, Pr[Z i =1]= 16 9 ρ i, i =1,...,t. Set I 0 = {1 i t : Z i =1}. Sice ρ i 1/16 for all 1 i t, we have that Pr[Z i =1] 1 9 ad E[ I 0 ] t 9. By stadard large deviatio argumets, I 0 t/10 with probability 1 o( 1 t ). Thus Pr[X = s] = [ ] Pr[I 0 = I] Pr Y i = s = Pr[I 0 = I]Pr[B( I, 9/16) = s] I [t] I [t] Pr[ I 0 <t/10] + i I I t/10 Pr[I 0 = I]Pr[B( I, 9/16) = s]. Here B(, p) deotes the biomial radom variable with parameters ad p. From kow estimates o biomial radom variables, we obtai that Pr[B(r, 9 16 )=s] c r 4c t for every r t 10, where c>0is a absolute costat. Pluggig this estimate ito the iequality above, we get the claimed result. Lemma 3.. Let U [] be a fixed set of U = k vertices. The the probability that i G(, p), with p as defied above, the graph G[U] is a regular graph is at most ( ) c 1k k/ for some absolute costat c1 > 0. Proof. Fix the degree of regularity d of the regular subgraph G[U] (this ca be doe i ways). Let U = {u 1,u,...,u k }. We boud the probability that the iduced subgraph G[U] isd-regular as follows. Expose the edges of G[U] by first exposig the edges from u 1 to U {u 1 }, the from u to U {u 1,u }, etc. If vertex u i gets t i eighbors i {u 1,...,u i 1 }, the u i should have exactly d t i eighbors i {u i+1,...,u k0 }. Recall that all edge probabilities i G(, p) are betwee 1/16 ad 9/16. Thus Propositio 3.1 applies, ad the probability of the latter evet is at most c 0 / k i. Multiplyig these probabilities for i =1,...,k, we derive that the probability that U is a d-regular graph is at most k 1 i=1 c 0 k i = (c 0) k 1 (k 1)! ( c1 ) k/, k where c 1 > 0 is a absolute costat ad the last iequality follows by applyig the Stirlig formula. We ca ow complete the proof of Theorem 1.4. We boud the probability that i G(, p) there exists a set U of size U = k such that U =(u 1 < <u k/ l k = a< <u k k/ l k = b< <u k ) ad G[U] is a regular graph by cosiderig two possible cases depedig o the differece t = b a betwee a ad b. Case 1. t c k 3/, where c > 0 is a small positive costat to be determied later.

9 LARGE NEARLY REGULAR INDUCED SUBGRAPHS 1333 The probability that there exists such a U is at most ( ( ) (3) ) t ( c1 ) k/. k/ l k k k/ l k k Ideed, there are less tha ( k/ l k) ways to choose the vertices u 1,...,u k/ l k 1,u k k/ l k+1,...,u k, less tha ways to choose the vertices a ad b so that the differece betwee them, t, is at most c k 3/, ad less tha ( t k k/ l k) ways to choose the vertices u k/ l k+1,...,u k k/ l k 1. For each such choice, the probability that the iduced subgraph o {u 1,u,...,u k } is regular is at most ( ) c 1k k/, by Lemma 3.. A simple computatio shows that the expressio i (3) is (much) smaller tha, say, 1/ for a appropriate choice of c. Ideed, sice k ad ( t k k/ l k) ( t ) k (et/k) k, this expressio is at most 3 k 4k/ l k ( et k ) k (c1 k ) k/ 3 k e 5k c k c k/ 1 = 3 [4e 10 c c 1 ] k/, implyig the required estimate by choosig, for example, c = 1 3e 5 c 1. Case. t c k 3/. Let U =(u 1 < <u k/ l k = a< <u k k/ l k = b< <u k ). Deote the first (smallest) k/ vertices of U by U 1, ad the last k/ vertices by U. Observe that if G[U] is a regular graph, the the two iduced subgraphs G[U 1 ] ad G[U ] have the same average degree. This is highly improbable. Ideed, by the defiitio of G(, p) the probability of each pair iside U 1 to become a edge is strictly less tha the probability of each pair iside U to become a edge. I additio, sice b a = t, the probability that each pair i, j U 1 with i<j,i a is a edge is less tha the probability that each pair i,j U with i <j,j b is a edge by Ω(t/) this is because i this case, ( i )( j ) ( i )( j ) Ω ( i + j ) i j ( j ) i Ω Ω ( ) t. It follows that the expected umber of edges iside U exceeds that iside U 1 by Ω( k t l k ). By Cheroff s iequality (cf., e.g., [1, Appedix A]), we obtai that the probability that G[U 1 ] ad G[U ] have the same average degree i G(, p) is at most exp{ c 3 k t /( l k)} for some absolute costat c 3 > 0. Thus, recallig our assumptio o t, we derive that the probability that there is U as above for which G[U] is a regular graph is at most ( ) k e c 3 k t l k ( e k ) ( k e c 3 c k5 e l k = c 3 c k e k4 l k ) k. Fially, as k C 1/ l 3/4 ad l k = Θ(l ), we ca choose C>0 to be large eough so that the above expressio is (much) smaller tha 1/.

10 1334 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV Combiig the two cases, we coclude that the probability that for ay fixed k which is at least C 1/ l 3/4 our graph cotais a iduced regular subgraph o k vertices is smaller tha / < 1/, ad as there are less tha choices for k, this shows that with positive probability the graph cotais o such subgraph. This completes the proof of Theorem The early regular case. Proof of Propositio 1.5. We will prove that for every (large) there exists a graph G o vertices i which every K-early regular iduced subgraph has at most 7K/log vertices. Assume first that is of the form =(s + 1) s for a positive iteger s. Notice that s =(1 o(1)) log. Take a set V of vertices ad partitio it ito s +1 disjoit equally sized subsets V 0,...,V s, V i = s. Now we defie G as follows. For i =0,...,s the set V i spas s i disjoit cliques of size i each. There are o edges betwee the cliques iside V i ad o edges betwee distict subsets V i V j i G. Assume ow that a subset U V spas a graph G[U] satisfyig δ(g[u]) = d, Δ(G[U]) Kd. Observe that the degrees of all vertices from V i i G are i 1, ad thus if i 1 <d, the U V i =. Let ow i d+1. Sice Δ(G[U]) Kd, U has at most Kd+ 1 vertices i each clique spaed by V i, implyig U V i s i (Kd+ 1). Therefore, U = s U V i = i=0 s i= log (d+1) i: i d+1 U V i s i (Kd +1)< (Kd + 1) s log (d+1) +1 (Kd +1) s (Kd +1) = d +1 d +1 ( + o(1))k s +1 log, implyig the desired result. For ot of the form =(s + 1) s, choose a miimal s satisfyig (s + 1) s. Let =(s + 1) s. It is easy to verify that 3. Now we ca apply the above costructio to create a graph G o vertices i which every K-early iduced subgraph has at most ( + o(1))k / log 7K/log vertices, ad the take G to be a arbitrary iduced subgraph of G o exactly vertices. 4. Ope problems. The most itriguig ope problem is that of obtaiig a better estimate for f(, 1). I particular, the cojecture of Erdős, Fajtlowicz, ad Stato that f(, 1)/ l teds to ifiity as teds to ifiity remais ope. The values of f(, 1) for 17 have bee determied by the authors of [5] ad by McKay, ad these are ideed larger tha the bouds that follow from the correspodig Ramsey umbers. Our upper ad lower bouds for f(, c) for c>1 are also rather far from each other, ad it will be ice to better uderstad the behavior of this fuctio. Oe ca also study a variat of the problems cosidered here that deals with ot ecessarily iduced subgraphs. Of course, every graph cotais a regular subgraph o all vertices (the subgraph with o edges), ad hece i this case it is atural to look for regular or early regular subgraphs with a large umber of edges. For every two positive itegers, m with m ( ) ad a real c 1, let g(, m, c) deote the largest g so that every graph with vertices ad m edges cotais a (ot ecessarily

11 LARGE NEARLY REGULAR INDUCED SUBGRAPHS 1335 iduced) c-early regular subgraph with at least g edges. The problems of determiig or estimatig the behavior of this fuctio seems iterestig. Here we ca establish tighter estimates tha the oes obtaied for the iduced case. Cosider first the case c = 1. Sice the complete graph o vertices ca be covered by matchigs (ad by ( 1) for eve ), it follows that every graph with m edges cotais a matchig of size at least m/. This implies that g(, m, 1) m/. The star K 1, 1 shows that for some values of m ad this is essetially tight, ad that g(, m, 1) = 1 for all 1 m <. By a simple applicatio of Szemerédi s regularity lemma it ca be show (see [8]) that for every fixed p > 0 there is a δ = δ(p) > 0 so that g(, p, 1) δ. This boud was sigificatly improved by Rödl ad Wysocka [9], who proved that every graph with vertices ad p edges cotais a r-regular subgraph with r αp 3 for some positive costat α. For a larger costat c observe, first, that complete bipartite graphs show that for m, g(, m, c) O(c(m/) )=O(cd ), where d =m/ is the average degree of a graph with vertices ad m edges. Ideed, a complete bipartite graph K k, k with k / has average degree d =Θ(k). Every c-early regular subgraph i it has miimum degree at most k, ad hece maximum degree at most ck. Thus it caot have more tha k ck = ck edges. Therefore, for every fixed c>1 there exists some C = C(c) so that g(m,, c) C(m/) for all m>. We ca show that for c> this is tight, up to a costat factor; amely, for ay c> there is a b = b(c) > 0so that g(, m, c) b(m/) for all m>. For simplicity we preset the proof oly for c = 5; the proof for ay other c> is similar. Theorem 4.1. Let G =(V,E) be a graph with V = vertices, E = m> edges, ad average degree d = d(g) = m/. The G cotais a 5-early regular subgraph with at least d edges. 1 Proof. We apply the method of Pyber [7] together with a few extra twists. Clearly we may assume that d 6. First, omit from G repeatedly vertices of degree smaller tha d/, as log as there are such vertices. As this process ca oly icrease the average degree, it eds with a oempty graph G with miimum degree at least d/. Now take a spaig bipartite subgraph of G with the maximum umber of edges. It is easy to see ad well kow that the degree of every vertex i this bipartite subgraph is at least half its degree i G, givig a bipartite graph H with miimum degree at least d/4. Put H 1 = H. Let A ad B deote the two vertex classes of H, where A B. Let A 1 A be a oempty subset of A which satisfies N H1 (A 1 ) A 1 ad A 1 is miimal with respect to cotaimet (subject to the coditio above ad to beig oempty). Clearly there is such a A 1,as N H1 (A) A ad N H1 (v) d/4 > 1 for all v A. Put N H1 (A 1 )=B 1 ad ote that by the miimality of A 1, A 1 = B 1. By the miimality, agai, ad by Hall s theorem, there is a matchig M 1 saturatig A 1 ad B 1. Let H be the graph obtaied from H 1 by deletig all edges of M 1.Now let A A 1 be a oempty, miimal subset of A 1 satisfyig N H (A ) A. As before, there is such a set, as N H (A 1 ) N H1 (A 1 ) = A 1. The miimality shows, agai, that i fact N H (A ) = A, ad that there is a matchig M saturatig A ad N H (A )=B. Proceedig i this maer we defie a sequece of sets A d/4 A d/4 1 A A 1 A ad B d/4 B d/4 1 B B 1 B where A i = B i for all i, ad a sequece of pairwise edge-disjoit matchigs M d/4,...,m,m 1, where M i is a perfect matchig betwee A i ad B i. Note that, ideed,

12 1336 NOGA ALON, MICHAEL KRIVELEVICH, AND BENNY SUDAKOV this process does ot termiate before these d/4 phases, as iitially all degrees i H are at least d/4, ad with the omissio of each matchig the degrees drop by 1. For coveiece we assume, from ow o, that d is a power of (otherwise, simply cosider oly the first d /4 sets A i,b i ad matchigs M i, where d >d/ is the largest power of that does ot exceed d). Note that A d/8 >d/8, sice every vertex of B d/4 is icidet with a edge of each of the matchigs M i for d/8 i d/4, ad all these edges are icidet with vertices of A d/8. We cosider two possible cases. Case 1. For every i, 0 i log d 4, I this case, A d/ i+4 > A d/ i+3. A 1 > log d 3 A d/8 d 64, ad the matchig M 1 is a regular subgraph with more tha d /64 edges, supplyig the desired result (with room to spare). Case. There is a i, 0 i log d 4, such that A d/ i+4 A d/ i+3. I this case, take the miimum i for which this holds. The A d/ i+3 > i A d/8 i 3 d. Let H d d be the graph cosistig of the matchigs M i+4 j for j< d i+4. The i+3 vertices of H are all those saturated by the largest matchig amog those, amely, M. The the maximum degree i d H d is exactly i+4 (as every vertex of A i+4 d/ i+3 d has that degree), ad the average degree is at least half of that, sice each of the i+4 matchigs M j above is of size at least half that of the largest oe, which is spaig. As i H the degree of every vertex of A d/ i+3 is exactly d, ad the total umber i+4 of edges of H is at least d A d/ i+3 i+4 d i 3 d i+4 = 7 d. Thus, H is a graph with maximum degree that exceeds the average degree by a factor of at most K =. We ca ow apply Propositio.1 with K = ad α =0.4 to coclude that H cotais a K/α = 5-early regular subgraph with at least edges, completig the proof. K Kα K 4α E(H ) d Ackowledgmets. We thak Domigos Dellamoica ad Vojta Rödl for poitig a error i a early versio of the paper. REFERENCES [1] N. Alo ad J. H. Specer, The Probabilistic Method, d ed., Wiley, New York, 000. [] F. R. K. Chug ad R. L. Graham, Erdős o Graphs: His Legacy of Usolved Problems, A. K. Peters, Ltd., Wellesley, MA, 1998.

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