How many random edges make a dense hypergraph non-2-colorable?

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1 How many random edges make a dense hypergraph non--coorabe? Benny Sudakov Jan Vondrák Abstract We study a mode of random uniform hypergraphs, where a random instance is obtained by adding random edges to a arge hypergraph of a given density. The research on this mode for graphs has been started by Bohman et a. in [7], and continued in [8] and [16]. Here we obtain a tight bound on the number of random edges required to ensure non--coorabiity. We prove that for any k-uniform hypergraph with Ωn k ɛ ) edges, adding ωn kɛ/ ) random edges makes the hypergraph amost surey non--coorabe. This is essentiay tight, since there is a -coorabe hypergraph with Ωn k ɛ ) edges which amost surey remains -coorabe even after adding on kɛ/ ) random edges. 1 Introduction Research on random graphs and hypergraphs has a ong history with thousands of papers and two monographs by Boobás [9] and by Janson et a. [15] devoted to the subject and its diverse appications. In the cassica Erdős-Rényi mode [1], a random graph is generated by starting from an empty graph and then adding a certain number of random edges. More recenty, Bohman, Frieze and Martin [7] considered a generaized mode where one starts with a fixed graph G = V, E) and then inserts a coection R of additiona random edges. We denote the resuting random graph by G+R. The initia graph G can be regarded as given by an adversary, whie the random perturbation R represents noise or uncertainty, independent of the initia choice. This scenario is anaogous to the smoothed anaysis of agorithms proposed by Spieman and Teng [19], where an agorithm is assumed to run on the worst-case input, modified by a sma random perturbation. Usuay, one investigates monotone properties of random graphs or hypergraphs; i.e., properties which cannot be destroyed by adding more edges, ike the property of containing a certain fixed subgraph. Given a monotone property A of graphs on n vertices, we can ask what are the parameters for which a random graph has property A amost surey, i.e. with probabiity tending to 1 as the number of vertices n tends to infinity. In our setting, we start with a fixed hypergraph H and inquire how many random edges R we have to add so that H +R has property A amost surey. This question is too genera to get concrete and meaningfu resuts, vaid for a hypergraphs H. Therefore, rather than considering a competey arbitrary H, we start with a hypergraph from a certain natura cass. One such cass of graphs was considered in [7], where the authors anayze the question of how many Department of Mathematics, Princeton University, Princeton, NJ 085. E-mai: bsudakov@math.princeton.edu. Research supported in part by NSF grant DMS , USA-Israei BSF grant, and by an Afred P. Soan feowship. Department of Mathematics, Princeton University, Princeton, NJ 085. E-mai: jvondrak@math.princeton.edu. 1

2 random edges need to be added to a graph G of minima degree at east dn, 0 < d < 1, so that the resuting graph G + R is amost surey Hamitonian. Further properties of random graphs in this mode are expored in [8]. In [16], Kriveevich et a. considered a sighty more genera setting, in which one performs a sma random perturbation of a graph G with at east dn edges. The authors obtained tight resuts for the appearance of a fixed subgraph and for certain Ramsey properties in this mode. In the same paper, they aso considered random formuas obtained by adding random k-causes disjunctions of k iteras) to a fixed k-sat formua. Kriveevich et a. proved that for any formua with at east n k ɛ k-causes, adding ωn kɛ ) random causes of size k makes the formua amost surey unsatisfiabe. This is tight, since there is a k-sat formua with n k ɛ causes which amost surey remains satisfiabe after adding on kɛ ) random causes. A reated question, which was raised in [16], is to find a threshod for non--coorabiity of a random hypergraph obtained by adding random edges to a arge hypergraph of a given density. For an integer k, a k-uniform hypergraph is an ordered pair H = V, E), where V is a finite non-empty set, caed the set of vertices and E is a famiy of distinct k-subsets of V, caed the edges of H. A -cooring of a hypergraph H is a partition of its vertex set V into two coor casses so that no edge in E is monochromatic. A hypergraph which admits a -cooring is caed -coorabe. -coorabiity is one of the fundamenta properties of hypergraphs, which was first introduced and studied by Bernstein [6] in 1908 for infinite hypergraphs. -coorabiity in the finite setting, aso known as Property B a term coined by Erdős in reference to Bernstein), has been studied extensivey in the ast forty years see, e.g., [10, 11, 13, 5, 18]). Whie -coorabiity of graphs is we understood being equivaent to non-existence of odd cyces, for k-uniform hypergraphs with k 3 it is aready NP -compete to decide whether a -cooring exists [17]. Consequenty, there is no efficient characterization of -coorabe hypergraphs. The probem of -coorabiity of random k-uniform hypergraphs for k 3 was first studied by Aon and Spencer []. They proved that such hypergraphs with m = c k /k )n edges are amost surey -coorabe. This bound was improved ater by Achioptas et a. [1]. Recenty, the threshod for -coorabiity has been determined very precisey. In [] it was proved that the number of edges for which a random k-uniform hypergraph becomes amost surey non--coorabe is k 1 n O1))n. Interestingy, the threshod for non--coorabiity is roughy one haf of the threshod for k-sat. It has been shown in [3] that a formua with m random k-causes becomes amost surey unsatisfiabe for m = k n Ok))n. The two probems seem to be intimatey reated and it is natura to ask what is their reationship in the case of a random perturbation of a fixed instance. Reca that from [16] we know that for any k-sat formua with n k ɛ causes, adding ωn kɛ ) random causes makes it amost surey unsatisfiabe. In fact, the same proof yieds that for any k-uniform hypergraph H with n k ɛ edges, adding ωn kɛ ) random edges destroys -coorabiity amost surey. Nonetheess, it turns out that this is not the right answer. It is enough to use substantiay fewer random edges to destroy -coorabiity: roughy a square root of the number of random causes necessary to destroy satisfiabiity. The foowing is our main resut. Theorem 1.1 Let k,, ɛ 0 be fixed and et H be a -coorabe k-uniform hypergraph with

3 Ωn k ɛ ) edges. Then the hypergraph H obtained by adding to H a coection R of ω n ɛ/) random -tupes is amost surey non--coorabe. Observe that for ɛ /, the resut is easy. Regardess of the hypergraph H, it is we known that a coection of ωn) random -tupes on n vertices is amost surey non--coorabe. So we wi be ony interested in the case when ɛ < /. For such ɛ we obtain the foowing resut, which shows that the assertion of Theorem 1.1 is essentiay best possibe. Theorem 1. For fixed k, and 0 ɛ < /, there exists a -coorabe k-uniform hypergraph H with Ωn k ɛ ) edges such that its union with a coection R of o n ɛ/) random -tupes is amost surey -coorabe. The rest of this paper is organized as foows. In the next section we present an exampe of the hypergraph which proves Theorem 1.. In Section 3, we discuss some natura difficuties in proving Theorem 1.1 and describe how to dea with them in the case of bipartite graphs. This resut aso serves as a basis for induction which we use in Section to prove the genera case of -coorabe k-uniform hypergraphs. Remark 1.3 We have two aternative ways of adding random edges. Either we can sampe a random -tupe R times, each time uniformy and independenty from the set of a n ) -tupes. Or we can pick each -tupe randomy and independenty with probabiity p = R / n ). Since -coorabiity is a monotone property, it foows, as in Boobás [9], Theorem. and a simiar remark in [16], that if the resuting hypergraph is amost surey non--coorabe -coorabe) in one mode then this is true in the other mode as we. This observation can sometimes simpify our cacuations. Notation. Let H = V, E) be a k-uniform hypergraph. In the foowing, we use the notions of degree and neighborhood, generaizing their usua meaning in graph theory. For a vertex v V, we define its degree dv) to be the number of edges of H that contain v. More generay, for a subset of vertices A V, A < k, we define its degree da) = {e E : A e}. For a k 1)-tupe of vertices A, we define its neighborhood as NA) = {w V \ A : A {w} E}. Aso, for a k )-tupe of vertices A, we define its ink as ΓA) = {{u, v} V \ A : A {u, v} E}. Throughout the paper we wi systematicay omit foor and ceiing signs for the sake of carity of presentation. Aso, we use the notations a n = Θb n ), a n = Ob n ) or a n = Ωb n ) for a n, b n > 0 and n if there are absoute constants C 1 and C such that C 1 b n < a n < C b n, a n < C b n or a n > C 1 b n respectivey. The notation a n = ob n ) means that a n /b n 0 as n, and a n = ωb n ) means a n /b n. The parameters k,, ɛ are considered constant. The ower bound The foowing exampe proves Theorem 1. and shows that our main resut is essentiay best possibe. 3

4 Construction. Partition the set of vertices [n] into three disjoint subsets X, Y, Z where X = Y = n 1 ɛ/. Let H be a k-uniform hypergraph whose edge set consists of a k-tupes which have exacty one vertex in X, one vertex in Y and k vertices in Z. By construction, the number of edges in H is Θn k ɛ ). X = n 1 ɛ/ Y = n 1 ɛ/ Z = n n 1 ɛ/ Figure 1: Construction of the hypergraph H. Caim. Coor a the vertices in X by coor 1 and vertices in Y by coor. Note that no matter how we assign coors to the remaining vertices, this gives a proper -cooring of H. Let R be a set of o n ɛ/) random -tupes. Then amost surey we can -coor Z so that none of the -tupes in R is monochromatic, i.e., there exists a proper -cooring of H + R. To prove this caim we transform R into another random instance R that contains ony singe vertices with a fixed prescribed coor and edges of size two which must not be monochromatic. Foowing Remark 1.3 we can assume that R was obtained by choosing every -tupe in [n] randomy and independenty with probabiity p = o n ɛ/ ). First note that amost surey there is no -tupe in R whose vertices are a in X or in Y. Indeed, since X = Y = n 1 ɛ/, the probabiity that there is such an -tupe is at most n ) 1 ɛ/ p = o1). Aso, every -tupe in R which has vertices in both X and Y is aready -coored so we discard it. For every v Z we add it to R with prescribed coor 1 if there is a subset A of Y of size 1 such that A {v} R. Since ɛ < / 1, the probabiity of this event is p 1 = ) Y p = 1 n 1 ɛ/ 1 ) p n )1 ɛ/) p = o n 1+ɛ/) = o n 1/). Simiary, if there is a subset B of X of size 1 such that B {v} R then we add v to R with prescribed coor. The probabiity p of this event is aso o n 1/). Fix an ordering v 1 < v <... of a vertices in Z. For every pair of vertices u, w Z we add an edge {u, w} to R if there is an -tupe L R such that the two smaest vertices in L Z are u and

5 w. Since the number of such possibe -tupes is at most n ), and ɛ < /, the probabiity of this event is ) n ) p 3 p = O n p = o n ɛ/ ) = o n 1). Aso note that by definition a the above events are independent since they depend on disjoint sets of -tupes. By our construction, any -cooring of Z in which singetons in R get prescribed coors and no -edge is monochromatic gives a proper -cooring of R. Therefore, to compete the proof of Theorem 1., it is enough to prove the foowing simpe statement. Lemma.1 Let R be a random instance which is obtained as foows. For i = 1, we choose every vertex in [n] with probabiity p i = o n 1/) independenty for i = 1, ) and prescribe to it coor i. In addition we choose every pair of vertices to be an edge in R with probabiity p 3 = on 1 ). Then amost surey there exists a -cooring of [n] in which a singetons in R get prescribed coors and no edge is monochromatic. Proof. Let G be the graph formed by edges from R. The probabiity that there is a vertex with conficting prescribed coors is np 1 p = o1). The probabiity that G contains a cyce is at most n s=3 ns p s 3 = On3 p 3 3 ) = o1). Finay the probabiity that there exists a path between two vertices with any prescribed coor is aso bounded by n s=1 ) n p 1 + p ) n s 1 p s 3 = o np 1 + p ) ) = o1). Therefore amost surey no vertex gets prescribed conficting coors, every connected component of G is a tree and contains at most one vertex with prescribed coor. This immediatey impies the assertion of the emma, since every tree can be -coored, starting from the vertex with prescribed coor if any). 3 Bipartite graphs Now et s turn to Theorem 1.1. First, consider the case of k = =. Here, we caim that for any bipartite graph G with Ω n ɛ) edges, adding ωn ɛ ) random edges makes the graph amost surey non-bipartite. This wi foow quite easiy, since it turns out that amost surey we wi insert an edge inside one part of a bipartite connected component of G, creating an odd cyce see the proof of Proposition 3.1). However, with the more genera hypergraph case in mind, we are aso interested in a scenario where random -tupes are added to a bipartite graph, and >. Then we ask what is the probabiity that the resuting hypergraph is -coorabe i.e., no -edge and no -edge shoud be monochromatic). We prove the foowing specia case of Theorem 1.1. Proposition 3.1 Let, 0 ɛ < / and et G be a bipartite graph with Ω n ɛ) edges. Then the hypergraph H obtained by adding to G a coection R of ω n ɛ/) random -tupes is amost surey non--coorabe. 5

6 A 1 A A i B 1 B B i Figure : Components of the bipartite graph G. Proof. Let G have cn ɛ edges, c > 0 constant. Consider the connected components of G which are bipartite graphs on disjoint vertex sets A 1, B 1 ), A, B ),... see Figure ). Denote a i = A i, b i = B i and assume a i b i. The number of edges in each component is at most a i b i. Since the tota number of edges is cn ɛ, we have a i a i b i cn ɛ. Observe that for =, the number of pairs of vertices inside the sets {A i } is a i ) 1 cn ɛ n) c n ɛ, so a random edge ands inside one of these sets with probabiity at east c n ɛ. Consequenty, the probabiity that none of the ωn ɛ ) random edges ends up inside some A i is at most 1 c n ɛ ) ωnɛ) = o1). Thus amost surey, G + R contains an odd cyce. On the other hand, in the genera case we are adding ω n ɛ/) random -tupes, which might never end up inside any vertex set A i. The probabiity of hitting a specific A i is a i ) / n ) = O a i /n ). For exampe, if G has n ɛ components with a i = b i = n 1 ɛ, then this probabiity is at most O a i /n) = O n )ɛ). Hence we need ω n )ɛ) random -tupes to hit amost surey some A i. This suggests a difficuty with the attempt to pace a random -tupe in a set which is forced to be monochromatic by the origina graph. We have to aow ourseves more freedom and consider sets which are monochromatic ony under certain coorings. More specificay, under any cooring, each of the sets A i must be monochromatic and at east haf of these sets must have the same coor. We do not know a priori which of the sets A i wi share the same coor, yet we can estimate the probabiity that any of these configurations aows a feasibe cooring together with the random -tupes. First, it is convenient to assume that the sets have roughy equa size, in which case we have the foowing caim. Lemma 3. Suppose we have t disjoint subsets A 1,..., A t of [n] of size Θn 1 α ). Let α ɛ/, t = Ω n α ɛ/)) and et R be a coection of ω n ɛ/) random -tupes on [n]. Then the probabiity that R can be -coored in such a way that each A i is monochromatic is at most e ωt). Proof. Consider the t possibe coorings in which a A i are monochromatic. For each such cooring there is a set of indices I, I t/ such that the sets A i, i I share the same coor. Since A i are 6

7 disjoint we have i I A i c 1 tn 1 α for some c 1 > 0. The probabiity that one random -tupe fas inside this set is at east c 1 ) tn 1 α / n ) c tn α ) for some c > 0. Hence Pr [ i I A i contains no -tupe from R ] 1 c tn α ) ) ωn ɛ/ ) e ωt n α ɛ/)) = e ωt) where we used t = t t = t Ωn α ɛ/) ). Therefore, by the union bound over a choices of I, we get Pr [ I such that i I A i contains no -tupe from R ] t e ωt) = e ωt). In particuar, amost surey there is no -cooring of R in which a A i are monochromatic. Now we can finish the proof of Proposition 3.1 for 3. Reca that G has cn ɛ edges. Partition the components of G according to their size and et G s contain a the components with A i [ s 1, s ). If there is any A i of size at east n 1 ɛ/, we are done immediatey because one of the ω n ɛ/) random -tupes amost surey ends up in A i and this destroys -coorabiity. So we can assume that G s is nonempty ony for s 1 ɛ/) og n. If we can choose a subgraph G s with sufficienty many edges, then we can use Lemma 3. to finish the proof as foows. Suppose there is s 1 ɛ/) og n such that G s has m s c s n 1 ɛ/) edges. each component of G s has at most s edges, the number of components of G s is t s s m s = Ω s n 1 ɛ/)). We set s = n 1 α, α ɛ/, which means that t s = Ω n α ɛ/)). To summarize, we have Ω n α ɛ/)) disjoint sets A i of size Θn 1 α ), each of which must be monochromatic under any feasibe cooring. Thus we can appy Lemma 3. to concude that for H = G + R, amost surey there is no feasibe -cooring. Finay, suppose that for any s 1 ɛ/) og n, the number of edges in G s is m s < c s n 1 ɛ/) and G s is empty for s > 1 ɛ/) og n. Then the tota number of edges is 1 ɛ/) og n s=1 m s < c 1 ɛ/) og n s=1 < c n 1 ɛ/) 1 s n 1 ɛ/) n 1 ɛ/) < cn ɛ As in the ast inequaity, we used 3). This contradicts our assumption that G has cn ɛ edges. Proof of Theorem 1.1 In this section we dea with the genera case of a -coorabe k-uniform hypergraph H, to which we add a coection of random -tupes R. Our goa is to prove the main theorem which asserts that if H has Ω n k ɛ) edges then adding to it ω n ɛ/) random -tupes makes it amost surey non--coorabe. The proof wi proceed by induction on k. The base case when k = foows from Proposition 3.1, so we can assume that k 3 and that the resut hods for k 1. Since we have ω n ɛ/) random -tupes avaiabe, we can divide them into a constant number of batches, where each batch sti has ω n ɛ/) 7

8 -tupes. We wi use a separate batch for each step of the induction. We write R = R 1 R... R k where R j = ω n ɛ/) for each j. We proceed in a series of emmas which aow us to make simpifying assumptions, and eventuay finish the proof of the theorem. The high-eve structure of the proof is as foows. 1. If H k contains Ωn k ɛ ) edges through k 1)-tupes of degree greater than n 1 ɛ/, we can prove by induction that H k + R is amost surey non--coorabe. Lemma.1 takes care of this case.. If H k contains Ωn k ɛ ) edges through k )-tupes of degree greater than n ) ɛ, we can aso prove by induction that H k + R is a.s. non--coorabe. This is proved in Lemma.. Aso, a variant of this emma can be used to finish the proof for = and a k If neither of the first two cases appy, we can cean up our hypergraph Lemma.5 and Lemma.6) to obtain a near-reguar hypergraph H α. The hypergraph provided by these emmas satisfies the conditions of Lemma.9 which proves directy that H α + R is amost surey non--coorabe. Lemma.1 Let k 3, and et H k be a k-uniform hypergraph on n vertices with c 1 n k ɛ edges. Consider a k 1)-tupes A V H k ) with degree greater than n 1 ɛ/. If there are at east c 1 n k 1 ɛ such k 1)-tupes in H k then H k + R is amost surey non--coorabe. Proof. For each k 1)-tupe A of degree > n 1 ɛ/, the neighborhood NA) contains Ω n ɛ/) distinct -tupes. Therefore a random -tupe ands inside NA) with probabiity Ω n ɛ/). Consequenty, the probabiity that none of ω n ɛ/) random -tupes from R k ends up inside NA) is 1 Ωn ɛ/ ) ) ωn ɛ/ ) = o1). If we have t c 1 n k 1 ɛ such k 1)-tupes, then the expected number of them, whose neighborhood does not contain any -tupe in R k, is ot). Therefore, by Markov s inequaity, we get amost surey at east t c 1 8 n k 1 ɛ k 1)-tupes with an -edge in their neighborhood. Denote by H k 1 the k 1)-uniform hypergraph formed by these k 1)-tupes. By induction, we know that H k 1 + R R k 1 is amost surey non--coorabe. Hence for every -cooring respecting R 1... R k 1, there is a monochromatic k 1)-tupe A in H k 1. Without oss of generaity assume that a vertices in A are coored by coor 1. By definition, the neighborhood NA) contains an -edge L R k. Either L is monochromatic, or one of its vertices x is coored by 1 as we. But then A {x} is a monochromatic edge of H k. This impies that there is no feasibe -cooring for H k + R R k. Thus we ony need to treat the case where there are at most c 1 n k 1 ɛ k 1)-tupes with degree greater than n 1 ɛ/, therefore at most c 1 n k ɛ edges through such k 1)-tupes. We wi get rid of these high degrees by removing their edges and making a degrees of k 1)-tupes at most n 1 ɛ/. This woud aso impy a bound of n ɛ/ on the degrees of k )-tupes, etc. However, in the foowing we show that for k )-tupes we can assume an even stronger bound. More specificay, we prove that if we have many edges through k )-tupes whose degree is at east n ) ɛ, then we can proceed by induction. For this purpose, we first show the foowing. 8

9 Lemma. Let and et G be a graph on n vertices with n δ edges. Then G contains 1 6 n1 δ disjoint subsets of vertices F 1, F,... such that in every F j, each vertex v F j can be assigned a set of neighbors Xv) Nv) so that x v = Xv) 1 n1 δ, Xv) Xw) = for v, w F j, and x v n )δ +. v F j We construct the sets F 1, F,... by a simpe agorithm. First, we show how to construct one such set for =. Caim.3 Assume that G has at most n vertices and at east 1 n δ edges. Then G contains a set of vertices F such that We can assign disjoint sets of neighbors Xv) Nv) to vertices v F, so that x v = Xv) 1 n1 δ and x v 1 16 n δ. v F The number of edges in G incident to F is at most 3n. Proof. We find F by the foowing procedure. We start with F = and add vertices one by one. We denote by NF ) = v F Nv) the vertices connected by an edge to F and by W = V \ F NF )) the remaining vertices. For a set S, we denote by d S v) the number of neighbors that vertex v has in S. Intuitivey, we choose a vertex from W which has a arge neighborhood but not overapping very much with NF ). Note that no vertex v W has neighbors in F, otherwise v woud be in NF ) itsef. Hence, by our construction, F is an independent set. We repeat the foowing, unti u F x u 1 16 n δ : 1. Find a vertex u W maximizing z u = d W u) 1 v Nu) W d W v) 1 n1 δ d NF ) u).. Set Xu) = Nu) W, x u = Xu) = d W u) and incude u in F. 3. Update W = V \ F NF )). We caim that as ong as u F x u < 1 16 n δ, we can aways find a vertex with z u = Ωn δ ) 0. Assuming this has been true up to a certain point, we have been choosing vertices with z u 0 and therefore v Nu) W d W v) d W u) = x u for each vertex when it was chosen. By incuding u in F, we increase the number of edges incident to NF ) by v Nu) W d W v) x u. Therefore, as ong as u F x u < 1 16 n δ, there are at most 1 8 n δ edges incident to NF ), and at east 3 8 n δ edges disjoint from NF ). These edges cannot touch F either or ese they woud touch NF )), so they are in the subgraph induced by W. 9

10 Now consider the choice of the next vertex u. Summing up z u over a avaiabe vertices, we get z u = d W u) 1 d W v) 1 n1 δ d NF ) u) v Nu) W = d W u) 1 d W v) 1 n1 δ enf ), W ) v W = 1 d W u) 1 n1 δ enf ), W ) where enf ), W ) denotes the number of edges between NF ) and W. As we mentioned above, the number of edges induced by W is at east 3 8 n δ which means that v W d W v) 3 n δ. By Cauchy-Schwartz, ) d W u) 1 d W u) 1 ) 3 W n n δ = 9 16 n3 δ. Aso, the number of edges incident to NF ) is bounded by 1 8 n δ. So we get z u 1 d W u) 1 n1 δ enf ), W ) 9 3 n3 δ 1 n1 δ 1 8 n δ = 1 n3 δ. Thus, there must be a vertex u W such that z u 1 n δ. Consequenty, we aso have x u z u 1 n1 δ. At the point we stop, we have u F x u 1 16 n δ, as required. Finay, we verify the number of edges incident to F. We distinguish two kinds of such edges. When we incude a vertex u in F, ca the edges connecting u to NF ) red, and ca the other edges connecting it to W bue. There can be at most n bue edges, because their endpoints form the disjoint sets Xu). The number of red edges can be bounded in the foowing way. Whenever we incude a vertex u in F, we have z u d W u) 1 n1 δ d NF ) u) 0. Therefore, the number of red edges contributed by vertex u at the moment when it is incuded in F is d NF ) u) d W u)/n1 δ = x u /n1 δ. Let v be the ast vertex which we added to F. By our construction, we have that u F \{v } x u 1 16 n δ. Therefore the tota number of red edges contributed by the vertices in F \ {v } is u F \{v } d NF ) u) u F \{v } x u n 1 δ < n 1 δ 1 16 n δ = n. The ast vertex v can possiby contribute at most n red edges. So the tota number of red and bue edges is at most n + n/ + n < 3n. Proof of Lemma.. Given the above caim, the proof of the emma foows easiy. We start with a graph G 1 = G. We iterate the construction of one set F, for j = 1,,..., 1 6 n1 δ. We appy the caim to the graph G j and find a set F j as required. Then we remove F j from the graph, to obtain G j+1. Since F j is incident to at most 3n edges, we can iterate up to 1 6 n1 δ times, and G j+1 sti contains at east 1 n δ edges. Each set F j satisfies u F j x u 1 16 n δ. Since x u 1 n1 δ, for 10

11 this impies u F j x u ) 1 n1 δ x u n )δ + u F j Lemma. Let k 3, and et H k be a k-uniform hypergraph on n vertices with c 1 n k ɛ edges. Consider k )-tupes of degree at east n ) ɛ. If there are at east c 1 n k ɛ edges through such k )-tupes then H k + R is amost surey non--coorabe. Proof. Consider a k )-tupe A of degree n δ, where δ ) ɛ. The ink of A in H k is a graph ΓA) with n δ edges. By Lemma., we find 1 6 n1 δ disjoint sets F j such that the vertices in each F j have disjoint sets of neighbors Xv) in ΓA), with sizes x v = Xv) satisfying v F j x v n )δ / +. We repeat this construction for each k )-tupe of degree D n ) ɛ. Note that for each such k )-tupe of degree D, we construct D/6n) sets as above. The sum of degrees of such k )-tupes is at east the tota number of edges through them, which is by assumption at east c 1 n k ɛ. Therefore, we obtain at east c 1 nk 1 ɛ sets F j in tota. Now consider a set F j chosen for a k )-tupe A. Ca it good if after adding the random -tupes in R k, there is at east one vertex in F j whose neighborhood in ΓA) contains a random -tupe. If this is not the case, ca it bad. We estimate the probabiity that F j is bad. By Lemma., the tota number of -tupes inside the sets Xv) for v F j is ) xv = Ω ) x v n )δ = Ω! + = Ω n ɛ/),! v Fj v F j where we used that δ ) ɛ and is a constant. Thus the probabiity that a random -tupe fas inside Xv) for some v F j is xv ) v F j / n ) = Ω n ɛ/ ). After adding the entire batch of random -tupes R k, Pr [ F j is bad ] = Consequenty, the expected fraction of bad F j s is o1). 1 Ω n ɛ/)) ωn ɛ/ ) = o1). By Markov s inequaity, this fraction is amost surey at most one haf, which means that at east c 1 8 nk 1 ɛ sets F j have a vertex v F j whose neighborhood contains some -tupe from R k. By the construction of the sets F j, for each one we have a set A of size k which together with v forms a k 1)-tupe B = A {v}. Since the F j s for a given k )-tupe A are disjoint, we obtain distinct pairs A, v) which correspond to distinct k 1)-tupes with a marked vertex v. We coud obtain the same k 1)-tupe B = A {v} c in k 1 different ways, but in any case we have at east 1 8k 1) nk 1 ɛ k 1)-tupes such that in H k, the neighborhood of each of them contains an -tupe from R k. Ca the hypergraph of these k 1)-tupes H k 1. By the induction hypothesis, H k 1 + R R k 1 is amost surey non--coorabe. Therefore, for any -cooring which respects the -edges from R R k 1, there must be a monochromatic k 1)-edge B in H k 1. However, since there is an -edge from R k in the neighborhood of B, one of its vertices shoud have the same coor as B. This forms a monochromatic edge in H k so there is no feasibe -cooring for H k + R R k. 11

12 Remark. Lemma. assumes that there are Ωn k ɛ ) edges through k )-tupes of degree at east n ) ɛ. However, one can easiy check that constant factors are not significant in the proof. In particuar, if H k is a k-uniform hypergraph on n vertices with c 1 n k ɛ edges such that there are at east c 1 n k ɛ edges through k )-tupes of degree at east 1 c 1n ) ɛ then H k + R is amost surey non--coorabe. Observe that in the case of =, there are aways at east 1 c 1n k ɛ edges through k )-tupes of degree at east 1 c 1n ) ɛ = 1 c 1n ɛ the remaining k )-tupes can contribute at most n 1 k ) c 1n ɛ 1 c 1n k ɛ edges). Therefore in the case of = we can aready concude that H k + R is amost surey non--coorabe. In the foowing, we can assume that 3 and at most c 1 n k ɛ edges go through k )-tupes of degree greater than n ) ɛ. Reca that from before, we can aso assume that at most c 1 n k ɛ edges go through k 1)-tupes of degree greater than n 1 ɛ/. In the foowing step, we remove these edges so that the degrees in the hypergraph are bounded. We aso make the hypergraph k-partite as described beow. Lemma.5 Let k, 3 and et H k = V, E) be a k-uniform hypergraph with c 1 n k ɛ edges, such that at most c 1 n k ɛ edges go through k 1)-tupes of degree n 1 ɛ/ and at most c 1 n k ɛ edges go through k )-tupes of degree n ) ɛ. Then H k contains a subhypergraph H k with the foowing properties 1. H k is k-partite, i.e. V can be partitioned into V 1 V... V k so that every edge of H k intersects each V i in one vertex.. Every vertex has degree at most n k 1 ) ɛ. 3. The degree of every k 1)-tupe is at most n 1 ɛ/.. The number of edges in H k is at east c n k ɛ, c = k! k k c 1. Proof. First, remove a edges through k 1)-tupes of degree greater than n 1 ɛ/ and through k )-tupes of degree greater than n ) ɛ. We get a hypergraph such that the degrees of a k 1)-tupes are at most n 1 ɛ/ and the degrees of a k )-tupes are at most n ) ɛ Consequenty, the degree of every vertex is at most n k 3 n ) ɛ = n k 1 ) ɛ. The number of remaining edges is at east 1 c 1n k ɛ. Next, we use a we-known fact, proved by Erdős and Keitman [1]. Every k-uniform hypergraph with m edges contains a k-partite subhypergraph with at east k! k k m edges. This can be achieved for exampe by taking a random partition V = V 1 V... V k and computing the expected number of edges which intersect each V i exacty once.) Let H k denote such a k-partite subhypergraph of H k. Its number of edges is at east c n k ɛ where c = k! c k k 1. Before the ast part of the proof, we make further restrictions on the degree bounds and structure of our hypergraph, by finding a subhypergraph H α with roughy reguar k 1)-degrees and sufficienty many edges. The number of edges that we can guarantee here is no onger a constant fraction of n k ɛ. The statement of Lemma.6 may appear technica but it is exacty what we need for our fina construction which finishes the proof Lemma.9). 1

13 Lemma.6 Let k, 3 and et H k be a k-uniform k-partite hypergraph on vertices V 1 V... V k with c n k ɛ edges, where the degrees of k 1)-tupes are bounded by n 1 ɛ/. Then H k contains a subhypergraph H α, α ɛ/, such that 1. The degree of every k 1)-tupe in V 1 V... V k 1 is either 0 or between n 1 α and n 1 α.. For some constant c 3 = c 3 k,, c ), the number of edges in H α is at east ɛ α k ɛ c 3 n + n k ɛ α ɛ/)). Proof. Consider a k 1)-tupes in V 1 V... V k 1 whose degree in H k is ess than 1 c n 1 ɛ. Deete a the edges through such k 1)-tupes, which is at most n k 1) 1 c n 1 ɛ 1 c n k ɛ edges in tota. We sti have at east 1 c n k ɛ edges eft. Now the degree of every k 1)-tupe in V 1 V... V k 1 is either 0 or between 1 c n 1 ɛ and n 1 ɛ/. We use an eementary counting argument to find the subhypergraph H α as required. Let n 1 α = s and partition V 1 V... V k 1 into groups of k 1)-tupes with degrees in intervas [ s, s+1 ), with s ranging between s 1 = og 1 c n 1 ɛ ) and s = og n 1 ɛ/ ). Let m s denote the number of edges through k 1)-tupes with degrees between s = n 1 α and s+1 = n 1 α. We prove the emma with c 3 = 1 16 c min { c 1/), 1 }. Assume for the sake of contradiction that m s < ɛ α k ɛ c 3 n + n k ɛ α ɛ/)) for every s, i.e. m s < 1 16 c n k ɛ c 1/) 1 n ɛ α Taking a sum from s = s 1 to s, we get and s s=s 1 ) + n α ɛ/) = 1 16 c n k ɛ c 1/) n 1 ɛ s + 1 s s1 s s i=s s 1 = 1 c n 1 ɛ ) = n 1 ɛ/) 1 1) c 1/) n 1 ɛ n 1 ɛ/). s n 1 ɛ/) Substituting into 1), we see that then the tota number of edges woud be s s=s 1 m s < 1 c n k ɛ which is a contradiction. Note that in this emma, we ose more than a constant fraction of edges. However, from now on, we do not use induction anymore and wi prove directy that H α + R is amost surey non--coorabe. We wi proceed in t = c 3 k n α ɛ/) stages. For each stage, we aocate a certain number of random -tupes. Namey, we set again R = R 1 R... R k, R j = ω n ɛ/). Furthermore, we divide each R j for j k 1 into t parts R j,1,..., R j,t so that ) n ɛ/ R j,i = ω = ω n ɛ/ α ɛ/)). t The random set R j,i wi be used for the j-th eve of the i-th stage. The foowing emma describes one stage of the construction. Finay, R k wi be used in the ast step of the proof. 13 ). 1)

14 Lemma.7 Let k, 3 and et H α be a k-uniform k-partite hypergraph where the degree of every k 1)-tupe in V 1 V... V k 1 is either zero or is in the interva [n 1 α, n 1 α ], and the number of edges in H α is at east k ɛ c 3 n α ɛ/). Then after adding sets of random edges R 1,i + R,i R k 1,i where R j,i = ω n ɛ/ α ɛ/)), there exists amost surey a famiy of q = k sets S 1,..., S q V k, n 1 α S i n 1 α, such that for every feasibe -cooring of H α + R 1,i R k 1,i, at east one S i is monochromatic. Proof. We are going to construct an -ary tree T of depth k 1. We denote vertices on the j-th eve by v a1 a...a j 1 where a i {1,,..., }. T is rooted at a vertex in V 1 and the j-th eve is contained in V j. We construct T in such a way that the vertices aong every path which starts at the root and has ength k 1 form a k 1)-tupe with degree Θn 1 α ) in H α. The neighborhoods of a branches of ength k 1 wi be our sets S i not necessariy disjoint). In addition, the set of chidren of every node on each eve j k, ike {v a1 a...a j 1 1, v a1 a...a j 1,..., v a1 a...a j 1 }, wi form an edge of R j,i. V 1 V V V 3 S 1 S R 1,i R,i S 3 S 9 Figure 3: Construction of the tree T, for k = and = 3. Branches of the tree form active k 1)-tupes, with neighborhoods S i. Each set of chidren on eve j + 1 forms an edge of R j,i. Assuming the existence of such a tree, consider any -cooring of H α + R 1,i R k 1,i. Since the chidren of each vertex on eve j < k 1 form an -edge in R j,i, every vertex has chidren of both coors. In particuar, there is aways one chid with the same coor as its parent. Therefore, starting from the root, we can aways find a monochromatic branch A of ength k 1. Since a the extensions of this branch to edges of H α must be -coored, a the vertices in S i = NA) must have the same coor. We grow the tree eve by eve, maintaining the property that a branches have sufficienty many extensions to edges of H α. More precisey, we ca an r-tupe in V 1... V r active if its degree is at east r = c 3 r nk r ɛ α ɛ/). 1

15 Caim.8 Every active r-tupe A, r k, can be extended to at east active r + 1)-tupes A {x}, x V r+1. r d r = n k r 1 α = c 3 1 n1 ɛ/+ α ɛ/) r+ Proof. Suppose that fewer than d r extensions of A are active. Since the degrees of k 1)-tupes are at most n 1 α, we get that any r + 1)-tupe has degree at most n k r 1 α. Therefore the number of edges through a active extensions of A is smaer than d r n k r 1 α = 1 r. We aso have inactive extensions of A which have degrees ess than r+1. The tota number of edges through these extensions of A is smaer than n r+1 = 1 r. But the tota number of edges through A is at east r. This contradiction proves the caim. We start our construction from an active vertex v V 1. Since H α has at east n 1 edges, such a vertex must exist. By our caim, v can be extended to at east d 1 active pairs {v, x}, x W V. Consider this set of d 1 vertices W. The probabiity that a random -tupe fas inside W is d1 ) / n ) = Ωn ɛ/+ α ɛ/) ). Now we use ωn ɛ/ α ɛ/) ) random -tupes from R 1,i that we aocated for the first eve of this construction. This means that amost surey, we get an -edge {v 1,..., v } R 1,i such that {v, v i } is an active pair for each i = 1,,...,. We continue growing the tree, using the random -tupes of R j,i on eve j. Since we have ensured that each path from the root to the eve j forms an active j-tupe, it has at east d j extensions to an active j + 1)-tupe. Again, the probabiity that a random -tupe hits the extension vertices W j+1 V j+1 for a given path is d j ) / n ) = Ω n ɛ/+ α ɛ/)). Amost surey, one of the -tupes in R j,i wi hit these extension vertices and we can extend this path to chidren on eve j + 1. The number of paths from the root to eve j is bounded by j 1 k which is a constant, so in fact we wi amost surey succeed to buid the entire eve. In this way, we amost surey buid the tree a the way to eve k 1. Every path from the root to one of the eaves forms an active k 1)-tupe and has degree [n 1 α, n 1 α ]. Define S 1, S,..., S q to be the neighborhoods of a these q = k paths. By construction, for any feasibe -cooring of H α + R 1,i R k 1,i, one of these paths is monochromatic which impies that the corresponding set S i is monochromatic as we. Lemma.9 Let k, 3 and et H α be a k-uniform k-partite hypergraph where the degree of every vertex is at most n k 1 ) ɛ, the degree of every k 1)-tupe in V 1 V... V k 1 is either zero or is in the interva [n 1 α, n 1 α ], and the number of edges in H α is at east ɛ α k ɛ c 3 n + k ɛ c3 n α ɛ/). Let R be a set of ωn ɛ/ ) random -tupes. Then amost surey, H α + R is not -coorabe. Proof. We appy Lemma.7 repeatedy in t = c 3 k n α ɛ/) stages. We partition the set R as described before into j=1 k 1 t i=1 R j,i R k, where R j,i = ω n ɛ/ α ɛ/)) and R k = ωn ɛ/ ). In each stage i, we amost surey obtain q = k sets S i,1,..., S i,q, n 1 α S i,j n 1 α such that for any -cooring of the hypergraph H α + R 1,i R k 1,i, one of these sets must be monochromatic. 15

16 If this happens, we ca such a stage successfu. After each successfu stage, we remove a edges of H α incident with any of the sets S i,1,..., S i,q. Since degrees are bounded by n k 1 ) ɛ and we repeat t = c 3 k n α ɛ/) times, the tota number of edges we remove is at most t i=1 j=1 q S i,j n k 1 ) ɛ tq n 1 α n k 1 ) ɛ = c 3 n ɛ α k ɛ ɛ α k ɛ c3 n. k ɛ In particuar, before every stage we sti have at east c 3 n α ɛ/) edges avaiabe, so we can use Lemma.7. Since the expected number of stages that are not successfu is ot), by Markov s inequaity, we amost surey get at east t/ successfu stages. Eventuay, we obtain sets S i,j for 1 i t/ and 1 j q such that For i 1 i and any j 1, j, S i1,j 1 S i,j =. For any i and any -cooring of H α + R 1,i + R,i R k 1,i, there is j i such that S i,ji monochromatic. is Finay, we add once again a coection R k of ωn ɛ/ ) random -tupes. We do not know a priori which seection of sets S i,j wi be monochromatic but there is ony exponentia number of choices q t/ = e Ot). For any specific choice of sets to be monochromatic, Lemma 3. says that the probabiity that after adding ωn ɛ/ ) random -tupes, there is a feasibe -cooring keeping these sets monochromatic, is e ωt). By the union bound, the probabiity that there exists a proper -cooring of H α + i,j R j,i + R k is at most q t/ e ωt) = o1). This competes the proof of Theorem 1.1, as outined at the beginning of Section. Acknowedgment. The first author woud ike to thank Michae Kriveevich for hepfu and stimuating discussions. We woud aso ike to thank both referees for hepfu remarks. In particuar, these remarks heped us notice a mistake in the origina proof of Lemma.. References [1] D. Achioptas, J.H. Kim, M. Kriveevich and P. Tetai, Two-cooring random hypergraphs, Random Structures and Agorithms 0 00), [] D. Achioptas and C. Moore, On the -coorabiity of random hypergraphs, in: Randomization and approximation techniques in computer science, Lecture Notes in Comput. Sci. 83, Springer, Berin, 00, [3] D. Achioptas and Y.Peres, The Threshod for Random k-sat is k og Ok), Journa of the AMS 17 00), [] N. Aon and J. Spencer, A note on cooring random k-sets, unpubished manuscript. [5] J. Beck, On 3-chromatic hypergraphs, Discrete Math. 1978),

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