Biased Positional Games and Small Hypergraphs with Large Covers

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1 Biased Positioal Games ad Small Hypergraphs with Large Covers Michael Krivelevich Tibor Szabó Submitted: Feb 4, 007; Accepted: Mar 3, 008 Mathematics Subject Classificatio: 91A43, 91A46, 05C65 Abstract We prove that i the biased (1 : b) Hamiltoicity ad k-coectivity Maker-Breaker games (k > 0 is a costat), played o the edges of the complete graph K, Maker has a wiig strategy for b (log o(1))/ log. Also, i the biased (1 : b) Avoider- Eforcer game played o E(K ), Eforcer ca force Avoider to create a Hamilto cycle whe b (1 o(1))/ log. These results are proved usig a ew approach, relyig o the existece of hypergraphs with few edges ad large coverig umber. 1 Itroductio Positioal games ivolve two players who alterately occupy the elemets of a give set V, the board of the game. The focus of their attetio is a give family F = {e 1,..., e k } V subsets of V, usually called the family of wiig sets. The players exchage turs occupyig oe previously uoccupied elemet of V. The game eds whe there are o uoccupied elemets of V. The game is specified completely by defiig who wis i every fial positio or, more geerally, for every possible game sceario. A iterested reader is ivited to read several survey papers of József Beck o the subject (see, e.g., [4]), ad also his forthcomig book [5] for a extesive backgroud o this fasciatig subject. 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 a USA-Israel BSF grat, by a grat from the Israel Sciece Foudatio ad by the Pazy Memorial Award. Istitute of Theoretical Computer Sciece, ETH Zurich, CH-809 Switzerlad. szabo@if.ethz.ch. of the electroic joural of combiatorics 13 (006), #R00 1

2 There are several types of positioal games depedig o how the idetity of the wier is determied. I this paper we restrict our attetio to two of them. 1.1 Maker-Breaker games I a Maker-Breaker game the first player, called Maker, wis if he completely occupies oe of the wiig sets by the ed of the game; the secod player, called Breaker, wis otherwise, i.e., if he maages to occupy at least oe elemet of (i.e., to break ito ) every wiig set by the ed of the game. Chvátal ad Erdős [8] were the first to explicitly cosider Maker-Breaker positioal games which are played o the edge set of the complete graph K, ad the wiig sets are defied by some graph theoretic property like coectivity or Hamiltoicity. I the coectivity game Maker wis if he creates a spaig tree by the ed of the game. I the Hamiltoicity game Maker wis if his graph cotais a Hamilto cycle i the ed. It seems that Maker, partly because he has so may (i.e., 1 ( ) ) edges by the ed, should be able to wi both of these games easily. This is ideed the case; for the coectivity game the wiig strategy is a triviality, for the Hamiltoicity game it requires a oe-page argumet [8]. Motivated by the easy success of Maker, Chvátal ad Erdős suggested to make the game more balaced by itroducig a bias: at each tur Breaker is allowed to occupy b edges istead of just oe, where b 1 is a iteger. I these games, as well as i other (1 : b) games, the most atural questio is, for a give value of the bias b, to determie which of the two players wis, assumig of course they play their optimal strategies. Chvátal ad Erdős proved [8] that if b < ( 1 o(1)) the Maker ca still occupy a spaig tree ad thus wi the coectivity 4 log game. They also proved [8] that the order of magitude of the bias is best possible. I fact they showed that if b > (1 + o(1)) the Breaker ca occupy all edges icidet to some vertex, log so Maker loses the coectivity game, sice his graph is discoected. Later Beck [1] improved the costat factor i the result of Chvátal ad Erdős ad established that Maker wis the coectivity game eve if b < (log o(1)) log. For the Hamiltoicity game Chvátal ad Erdős cojectured that there is fuctio b H () tedig to ifiity such that Maker ca still build a Hamilto cycle if he plays agaist a bias b H (). Their cojecture was verified by Bollobás ad Papaioaou [7] who proved that Maker is able to build a Hamilto cycle eve if Breaker s bias is as large as c log. Beck improved greatly log log o this [] ad established that the order of magitude of the critical bias is the same for the Hamiltoicity game ad the coectivity game. He showed that Maker wis the Hamiltoicity game provided Breaker s bias is at most ( log o(1)). 7 log I this paper we improve the costat factor i the Hamiltoicity game result of Beck ad the electroic joural of combiatorics 13 (006), #R00

3 achieve the same lower boud as is kow for the coectivity game. Theorem 1 Maker wis the (1 : b) Hamiltoicity game for every b < (log o(1)). log Our proof techique provides the same lower boud for the critical bias i the k-coectivity game, where the family C k of wiig sets cosists of the edgesets of k-coected spaig graphs. Theorem Maker wis the (1 : b) k-coectivity game for every b < (log o(1)) 1. Avoider-Eforcer games log. We also study the misère versio of Maker-Breaker games. I a Avoider-Eforcer game the first player, called Avoider, wis if he does ot occupy completely ay of the wiig sets by the ed of the game; otherwise the secod player, called Eforcer, wis. Although Avoider-Eforcer games appear aturally i several situatio related to Maker- Breaker games, our uderstadig of them is much less satisfactory. The Avoider-Eforcer versios of graph games ofte behave surprisig differetly from their Maker-Breaker couterparts. For example Hefetz ad the authors [13] recetly proved that i the (1 : b) Avoider-Eforcer coectivity game the critical bias is liear i : Avoider wis if ad oly if the bias of Eforcer is at least (1 + o(1))/. This is i strikig cotrast with the order log of the critical bias i the Maker-Breaker coectivity game. The Avoider-Eforcer Hamiltoicity ( ) game was also aalyzed i [13], where it was prove that if the bias of Eforcer is O the Eforcer log (4) log log (3) wis (the otatio log (k) stads for the k times iterated logarithm of ). Here we improve this result ad obtai the same order of magitude as for the Maker-Breaker games with a better costat. The theorem resolves oe of the seve most humiliatig ope problems listed i the book of Beck [5]. Theorem 3 Eforcer wis the (1 : b) Hamiltoicity game for every b < (1 o(1)). log A major differece we have here compared to the Maker-Breaker couterpart is the complete lack of results from the other directio. While we kow the order of magitude of the critical bias i the Maker-Breaker Hamiltoicity game, for Avoider s wi the best available strategy is the trivial oe: Avoider (clearly) wis if he has less the edges by the ed of the game (which happes for bias b = (1 + o(1))/). games. Similarly to Maker-Breaker games, our techique carries the boud through to k-coectivity the electroic joural of combiatorics 13 (006), #R00 3

4 Theorem 4 Eforcer wis the (1 : b) k-coectivity game for every b < (1 o(1)) log. To the best of our kowledge the k-coectivity game (i both the Maker-Breaker ad Avoider- Eforcer versios) has ot bee cosidered explicitly i the literature for k. 1.3 Notatio ad Termiology A hypergraph F is a subset of the power set V where V is a fiite o-empty set (vertex set). The members of F are called hyperedges. We say that a positioal game is played o a hypergraph F where V is the board of the game, ad F is the collectio of wiig sets. The idetity of the player makig the first move does ot affect our asymptotic results; uless otherwise stated, we set as a default that Maker or Avoider starts the games. If all hyperedges of a hypergraph F are of size r, the F is called r-uiform. A subset T V is a cover of F if T e for every e F. The coverig umber of F, deoted by τ(f), is the miimal cardiality of a cover of F. For a graph G ad a subset U V (G), we deote by N(U) the exteral eighborhood of U, i.e., N(U) = {v V (G) \ U : v has a eighbor i U}. Wheever eeded we assume that the uderlyig parameter is large eough. All logarithms are atural uless stated explicitly otherwise. We will igore systematically roudig ad divisibilty issues. The radom graph ituitio ad ope problems I this sectio we discuss a paradigm, poited out first by Chvátal ad Erdős [8] ad covered later i great detail i several papers of Beck (c.f. [3]). This paradigm relates positioal games to radom graphs. The goal of the sectio is twofold. O the oe had it describes this paradigm, called radom graph ituitio, which is a major drivig force behid our curret results. O the other had, we list here some of the ope problems we fid iterestig which are motivated by the radom graph ituitio. Biased positioal games, as defied above, are perfect iformatio games played by two players, ad as such they apparetly do ot leave much room for probabilistic cosideratios. Yet, as the accumulated research experiece covicigly shows, probabilistic ituitio is of utmost importace whe aalyzig biased games. For biased (1 : b) games, the most importat questio is who wis for a give value of bias b. It is ofte quite easy to see that for small eough b Maker wis, while whe b is large eough the wi is Breaker s. It is therefore atural to ask for the critical value of b at which the game chages hads ad becomes Breaker s wi the electroic joural of combiatorics 13 (006), #R00 4

5 istead of Maker s Oe ca easily prove formally that there is bias mootoicity, i.e., the game chages hads at most oce. More formally, for a give hypergraph F let b F be the largest iteger b for which Maker wis the (1 : b) game played o the hypergraph F. The iteger b F is called the critical bias of the positioal game F. It ca easily be proved formally that Maker wis the (1 : b) game for every b b F. Paul Erdős suggested the followig heuristic argumet to guess the critical bias. Durig the game Maker claims m (b+1) edges, the rest belogs to Breaker. Of course, both players are assumed to play perfectly. Yet, if istead each of them would play completely at radom, RadomMaker s graph would look like a radom graph with vertices ad m edges. model of radom graphs is usually deoted by G(, m). Erdős suggested to look at the critical m for which the correspodig combiatorial property (coectivity, Hamiltoicity etc.) starts appearig almost surely i G(, m) as a hit for the critical bias b. Sice the theory of radom graphs is a rather well developed field, where quite a few questios about the locatio of a threshold for the appearace of various combiatorial properties have bee solved successfully, Erdős paradigm allows for guessig the critical bias for a variety of biased positioal games. Ad ideed, for the coectivity ad Hamiltoicity games the results obtaied fit Erdős predictio surprisigly well. I order to covice the reader i this thesis, ad also as a future referece poit, we state kow results about coectivity ad Hamiltoicity i G(, m). It is well kow that, for ay costat k 1, if m = m() = (log + (k 1) log log ω()), the the radom graph G(, m) has almost surely a vertex of degree at most k 1 ad is thus ot k-coected. O the other had, if m = m() = (log + (k 1) log log + ω()), the G(, m) is almost surely k-coected [9]. (The term almost surely meas with probability tedig to 1 as the umber of vertices teds to ifiity ; ω() stads for ay fuctio tedig to ifiity abritrarily slowly with.) As for Hamiltoicity, applyig the above lower boud for k =, oe ca derive that G(, m) is almost surely o-hamiltoia for m = m() = (log + log log ω()). Komlós ad Szemerédi [15] ad idepedetly Bollobás [6] proved that if m = m() = (log + log log + ω()) the G(, m) is almost surely Hamiltoia. The above statemets suggest that the critical bias b for both the coectivity ad Hamiltoicity games should be of order / log, ad this is ideed matched by available results o positioal games. (Oe should ote though that upo a closer ispectio the positioal games results are sizably less accurate tha their radom graphs couterparts.) The results of the curret paper uify the kow lower bouds for all the above games. Deote by b Dk, b Ck, b H, b T the critical biases for the miimum degree k, k-coectivity, Hamiltoicity ad coectivity games, respectively. We ow have that (log o(1)) log b H b T b D1 (1 + o(1)) log the electroic joural of combiatorics 13 (006), #R00 5 This

6 ad (log o(1)) log b C k b Dk b D1 (1 + o(1)) log. Hece the foremost obstacle stadig i the way of the asymptotic determiatio of the critical bias for the coectivity or Hamiltoicity game is the iability of our curret techiques to deal with the midegree-1 game. I other words, what is the smallest bias of Breaker which allows him to isolate a vertex i Maker s graph? Problem 1 Determie b D1 asymptotically. I the model of radom graph process a eve stroger coectio is true betwee, say, the properties of coectivity ad midegree-1. I a iformal laguage this says that the mai reaso a radom graph is ot coected is that there exists a isolated vertex. This motivates our questio whether a similar pheomeo holds i the theory of biased positioal games. We thik the aswer is yes, i.e., the oly reaso Maker caot wi the coectivity game is that Breaker is able to isolate a vertex i Maker s graph. Cojecture 1 b T = b D1. We are curious whether oe ca show aythig of this sort without obtaiig the asymptotic value of these critical biases. Other atural questios motivated by kow facts o radom graph processes are Is it true that b H = b D? Is it true that b Ck = b Dk? For Avoider-Eforcer games our kowledge of the critical bias is much more limited tha for the Maker-Breaker case. I fact the situatio is more complicated as Avoider-Eforcer games lack bias-mootoicity i geeral [13]. Still, at least for some of these games the above described Erdős paradigm ca serve as a very good guess i decidig the wier for a give value of bias b. We would like to close the ope problem sectio with a admittedly modest cojecture which just highlights our lack of uderstadig of Avoider-Eforcer games. Cojecture Prove that Avoider has a wiig strategy i the (1 : ) Hamiltoicity game. 10 the electroic joural of combiatorics 13 (006), #R00 6

7 3 Tools of the trade For our proofs oe eeds to exted the cocept of positioal games so that Maker ad Avoider ca also have a bias larger tha oe. Give a hypergraph F o a vertex set V, i a (p : q) positioal game the first player occupies p elemet of V i each roud, while the secod player occupies q i each roud. Oe of the most importat techical tools i the theory of biased positioal games is the followig criterio for Breaker s wi due to Beck [1]. If e F (1 + q) e /p < q, (1) the Breaker has a wiig strategy (irregardless of who moves first). Historically, sice Erdős ad Selfridge established the above criterio for the ubiased case p = q = 1 i [10], iequality (1) is ofte called the biased Erdős-Selfridge criterio. The Avoider-Eforcer versio of the above criterio has bee obtaied recetly i [13] (see also [5]). I a (p : q) biased Avoider-Eforcer game, played o a collectio F of wiig sets, Avoider wis provided e F ( 1 + p) 1 e ( < p. () p) Oe should ote that the above criterio does ot ivolve the parameter q at all ad is thus less powerful compared to the correspodig Maker-Breaker coditio (1) of Beck. Nevertheless, the latter criterio will be good eough for our purposes. 4 A example I this sectio we preset ad discuss a relatively simple example, whose purpose is to illustrate a stadard approach to aalyzig biased positioal games, its limitatios, ad also to idicate the mai idea behid our proof. Cosider the followig (1 : b) Maker-Breaker game played o the edges of the complete graph K. A additioal parameter k 1 is give (the reader should thik about k as a costat). Maker wis the midegree-k game if his graph has miimum degree at least k by the ed. The correspodig hypergraph, whose members are the edgesets of graphs with miimum degree k, is deoted by D k. Observe that accordig to the above cited Chvátal-Erdős result [8] whe b (1 + o(1))/ log, Breaker ca isolate a vertex ad thus wi the midegree-k game for every positive k. We will be iterested i fidig the bias b as large as possible for which the electroic joural of combiatorics 13 (006), #R00 7

8 Maker still wis, ad therefore i fidig good strategies ad criteria for Maker s wi i the midegree-k game. As is the case with quite a few combiatorial games, i order for Maker to achieve a wi, it turs out very useful to iterpret the game differetly ad to view Maker as playig Breaker s role i this differet iterpretatio. This approach was first suggested by Beck i [1] ad later used frequetly by him ad other authors. Its mai advatage is i the possibility of applyig criterio (1) for Breaker s wi i biased games. For each v V (K ), defie a hypergraph F v composed of all sets of k edges of K icidet to v. Clearly, F v is a ( k)-uiform hypergraph with E(F v ) = ( ) ( 1 k = 1 ) k 1 hyperedges. Observe that if Maker maages to occupy a edge (of K ) i every hyperedge of F v, the he has at least k edges icidet to v. (If this is ot the case, the Breaker has some 1 (k 1) = k edges icidet to v i his possessio, those edges form a hyperedge of F v completely occupied by Breaker.) Therefore, i order for Maker to wi the midegree-k game, it suffices to break ito every hyperedge of the hypergraph F = v V (K F ) v. The hypergraph F is ( k)-uiform with ( 1 k 1) edges. Applyig the biased Erdős-Selfridge criterio (1) to F ad substitutig p = b, q = 1, it follows that if ( ) 1 k b < 1 k 1, the Maker wis the midegree-k game. Stadard asymptotic maipulatios show that the largest b = b() for which the above iequality is valid is ( ) log b = + o(1) k log. Let us evaluate the obtaied result. First of all, it is ot that bad the lower boud for the critical bias we have just obtaied is of the same order of magitude as the Chvátal- Erdős upper boud ad the costat factor depeds oly liearly o k. However, recallig the ituitio comig from radom graphs such a result is ot satisfactory. Ideed, the critical umber of edges m, for which the radom graph G(, m) starts havig almost surely miimum degree at least k, depeds o k very weakly it is ( 1 + o(1)) log for ay costat k. Let us retur to our argumet. A simple observatio, playig a crucial role i the curret paper, is that the most importat property of the hypergraph F v was the fact that its coverig umber is at least k (actually, it is exactly k). If Maker hits every hyperedge of F v durig the game the his edges icidet to v form a cover of F v, ad, as τ(f v ) k, we ca coclude that Maker occupies at least k edges of K icidet to v. We ca coclude that istead of F v it is potetially beeficial to choose a hypergraph with fewer hyperedges of still relatively large size (to make it easier for Maker to hit each of them), the electroic joural of combiatorics 13 (006), #R00 8

9 yet with coverig umber large eough. Turig this paradigm ito a quatitative relatio betwee the parameters ivolved will play a key role i derivig the improved results. We are thus iterested i showig the existece of hypergraphs with few hyperedges ad large coverig umber. This is discussed i the ext sectio. 5 Large covers of small hypergraphs For positive itegers, r, t satisfyig r, ad t r + 1, defie f(, r, t) = mi{ F : F is a r-uiform hypergraph o vertices, τ(f) t}. Claim 1 For every s t, ( f, s t + 1 ), t s ( ) s. t 1 Proof. Partitio [] = s i=1 A i, A i = /s. Defie a hypergraph whose vertex set is V = [], ad whose hyperedge set is F = { i I A i : I [s], I = s t + 1}. Clearly F is a s t+1 uiform hypergraph. If a set T [] itersects less tha t of the sets s A i, the there is at least oe hyperedge of F left completely outside T, ad thus T is ot a cover of F. Therefore, τ(f) t. For compariso, here is a lower boud for f(, r, t). If F is a r-uiform hypergraph o vertices with τ(f) t, the every set U V, U = t 1, misses at least oe member of F, implyig: F ( t 1 ( r t 1 I case r = s t+1 ad s, t = o(), both lower ad upper bouds o f(, r, t) are of the form s (cs/(t 1)) t 1. ) ). 6 Biased Hamiltoicity game I this sectio we obtai a ew estimate for the critical bias i the Maker-Breaker Hamiltoicity game. This estimate improves substatially the curretly best kow result due to Beck [] ad comes quite close to the upper boud of Chvátal ad Erdős [8]. The proof of Theorem 1 is a combiatio of our basic thiig trick ad the followig Hamiltoicity criterio derived recetly by Hefetz ad the authors [14]. the electroic joural of combiatorics 13 (006), #R00 9

10 Lemma 1 Let 1 d e 3 log ad let G be a graph o vertices satisfyig properties P1, P below: P1 For every S V, if S k 1 (, d) := log log log d d log log log log the N(S) d S ; P There is a edge i G betwee ay two disjoit subsets A, B V such that A, B k (, d) := log log log d log log log log The G is Hamiltoia, for sufficietly large. I fact, coditios P1 ad P above will be quite hady i the k-coectivity game as well. Theorem 1 is ow a immediate corollary of Lemma 1 ad the followig theorem. Theorem 5 Let b = log + log log log log. I a biased (1 : b) Maker-Breaker game played o the edges of the complete graph K, Maker has a strategy to build a graph which satisfies properties P1 ad P with d = d() = log, provided is large eough. log log Remark. No real attempt is made here to optimize the error term i the above expressio for b. Proof. We itroduce some otatio: k1 = k 1 () = k (log log ) 3 1(, d) = (1 + o(1)) log log log log, k = k() = (log log ) k (, d) = (1 + o(1)) 4130 log log log log.. For every 1 k k1 ad for every set S [], S = k, let F(S) be a r(k)-uiform hypergraph o []\S with at most ( ) s(k) dk hyperedges whose coverig umber satisfies: τ(f(s)) dk + 1, where s(k) = k log, r(k) = log d ( ) 1 ( k) = 1 ( k). log log log The existece of such a hypergraph is guarateed by Claim 1. Observe that if, by the ed of the game, i Maker s graph M the set S is coected by a edge to every member of F(S), the N M (S) τ(f(s)) > dk. Ideed, i this case the exteral eighborhood of S i M forms a cover of F(S). It thus follows that i order to esure Coditio P1, it is eough to guaratee that i Maker s graph M, for every 1 k k 1 ad for every set S [], S = k, S is coected to every edge from F(S). I order to meet coditio P, it is eough to guaratee that i M the electroic joural of combiatorics 13 (006), #R00 10

11 there is a edge betwee every pair of disjoit sets A, B [] of size A = B = k. By the geeralized Erdős-Selfridge coditio (1) it is therefore eough to check that: or k=1 k 1 k=1 S =k k 1 ( )( ) k log k dk F(S) k r(k) b + k (1 1 log log )( k) b + A, B =k (k ) b < 1, (3) ( )( ) k k k (k ) b < 1. We will prove that both summads i the left had side of the iequality above are asymptotically egligible. The first oe ca be estimated from above by: = [ k 1 ( ) d e log (1 1 d k=1 [ k 1 k=1 k 1 k=1 log + log log log log (e log log ) log log log log [ O( log log ) ] k log log )(1 )(log k + log log log log ) log «] k 1 1 log log k 1 log 1 log log log log log log log ] k = o(1). The secod summad is at most: [ (e ) ] k [ (O(1) ) k log log log log Θ(1)(log log ) k b = (log log ) log log log ( O(1) log log (log log )1.9) k = o(1). ] k This fiishes the proof of Theorem 5. 7 Biased k-coectivity game I this sectio we prove Theorem. To this ed we show that if the bias b satisfies b = (log + o(1)) the Maker ca build a k-coected graph i the biased (1 : b) game, for log ay costat k > 0 (ad i fact for some slowly growig k as well). Observe that this weak depedece of the obtaied result o k matches very well our ituitio derived from radom graphs ad discussed i Sectio. the electroic joural of combiatorics 13 (006), #R00 11

12 Theorem 6 I a biased (1 : b) Maker-Breaker game, Maker has a strategy for creatig a log -coected graph provided log log ad is large eough. b = log + log log log log Remark. Here, too, we do ot make ay serious attempt to get the best possible coectivity result, our aim is rather to get k-coectivity for some k 1, whe the bias is b = (1 o(1))/ log. Proof. I Theorem 5 we showed that Maker ca create a graph M satisfyig coditios P1 ad P of Lemma 1 with d() = log. We claim that every graph M meetig these coditios with log log the above value of d() is d()-coected. Let S be a subset of V (M) of cardiality S < d(). Assume to the cotrary that S cuts G ito two o-empty pieces: V (M) \ S = A B so that A B ad M has o edges betwee A ad B. We will use the otatio of the proof of Theorem 5. First, observe that A k1 d d. Ideed, if A k 1, the by P1 we have: N(A) d A d ad N(A) S a cotradictio. If k1 A k1d d, choose A 0 A of cardiality A 0 = k 1, the by P1 we have N(A 0) d A 0 = k 1 d ad N(A 0) (A \ A 0 ) S, implyig a claimed boud o A. But the A k ad sice B A k, by Property P there is a edge of M betwee A ad B a cotradictio. Theorem implies immediately the followig cosequece for the midegree-k game cosidered i Sectio 4. Corollary 1 For every fixed k > 0 Makers wis the (1 : b) midegree-k game provided ad is sufficietly large. b log + log log log log,, 8 Biased Avoider-Eforcer games For biased Avoider-Eforcer games we ca use a very similar proof strategy to that of Theorem 5 to ifer the followig. Theorem 7 Let b log + log log log log. I a biased (1 : b) Avoider-Eforcer game, Eforcer has a strategy to force Avoider to create a graph satisfyig P1 ad ad P with d = d() = provided is large eough. log log log the electroic joural of combiatorics 13 (006), #R00 1

13 The oly major differece is that istead of the geeralized Erdős-Selfridge criterio (1), Eforcer uses criterio () to make sure that Avoider s graph satisfies properties P1 ad P. Pluggig ito Criterio (), iequality (3) ow becomes k 1 k=1 S =k ( F(S) ) k r(k) + ( ) (k ( ) < ) b. (4) b b b A, B =k We ca use ow the asymptotic estimate 1 + x = e x+θ(x) for x 0. The rest of the proof remais the same mutatis mutadis. To derive Theorem 3 we ca apply Lemma 1. For Theorem 4 we oly ote that, as proved i the previous sectio, properties P1 ad P for d() = log / log log imply d()-coectivity. Note added i proof. Very recetly, Gebauer ad the secod author proved [11] that the thresholds bias for both the coectivity game ad the midegree-k game are asymptotically equal to / log. Also, Hefetz, Stojaković ad the authors defied i [1] a ew ad rather atural versio of the Avoider-Eforcer game ad proved that i this versio the threshold bias for the Hamiltoicity game equals asymptotically to / log, where the lower boud essetially comes from the preset paper. Refereces [1] J. Beck, Remarks o positioal games, Acta Math. Acad. Sci. Hugar. 40 (198), [] J. Beck, Radom graphs ad positioal games o the complete graph, Aals of Discrete Math. 8 (1985) [3] J. Beck, Determiistic graph games ad a probabilistic ituitio, Combi., Prob. Computig 3 (1994), [4] J. Beck, Positioal games, Combi., Prob. Computig 14 (005), [5] J. Beck, Combiatorial Games, Cambridge Uiv. Press, 008. [6] B. Bollobás, The evolutio of radom graphs, i Graph Theory ad Combiatorics (Cambridge, 1983), Academic Press, Lodo, 1984, pp [7] B. Bollobás ad A. Papaioaou, A biased Hamiltoia game, Cogressus Numeratium, 35 (198), the electroic joural of combiatorics 13 (006), #R00 13

14 [8] V. Chvátal ad P. Erdős, Biased positioal games, Aals of Discrete Math. (1978), 1 8. [9] P. Erdős ad A. Réyi, O the stregth of coectedess of a radom graph, Acta Math. Acad. Sci. Hugar. 1 (1961), [10] P. Erdős ad J. Selfridge, O a combiatorial game, J. Comb. Th. Ser. A 14 (1973), [11] H. Gebauer ad T. Szabó, Asymptotic radom graph ituitio ad the biased coectivity game, submitted. [1] D. Hefetz, M. Krivelevich, M. Stojaković ad T. Szabó, Avoider-Eforcer: the rules of the game, submitted. [13] D. Hefetz, M. Krivelevich ad T. Szabó, Avoider-Eforcer games, J. Comb. Th. Ser. A 114 (007), [14] D. Hefetz, M. Krivelevich ad T. Szabó, Hamilto cycles i highly coected ad expadig graphs, Combiatorica, to appear. [15] J. Komlós ad E. Szemerédi, Limit distributios for the existece of Hamilto circuits i a radom graph, Discrete Math. 43 (1983), the electroic joural of combiatorics 13 (006), #R00 14