HAMBURGER BEITRÄGE ZUR MATHEMATIK

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1 HAMBURGER BEITRÄGE ZUR MATHEMATIK Heft 353 Extremal results for random discrete structures Mathias Schacht, Hamburg Version Oktober 2009

2 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES MATHIAS SCHACHT Abstract We study thresholds for extremal properties of random discrete structures We determine the threshold for Szemerédi s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turán-type problems for random graphs and hypergraphs In particular, we verify a conjecture of Kohayakawa, Luczak, and Rödl for Turán-type problems in random graphs Similar results were obtained by Conlon and Gowers 1 Introduction Extremal problems are widely studied in discrete mathematics Given a finite set Γ and a family F of subsets of Γ an extremal result asserts that any sufficiently large or dense subset G Γ must contain an element from F Often all elements of F have the same size, ie, F Γ k for some integer k, where Γ k denotes the family of all k-element subsets of Γ For example, if Γ n = [n] = {1,, n} and F n consists of all k-element subsets of [n] which form an arithmetic progression, then Szemerédi s celebrated theorem [40] asserts that every subset Y [n] with Y = Ωn contains an arithmetic progression of length k A well known result from graph theory, which fits this framework, is Turán s theorem [41] and its generalization due to Erdős and Stone [12] see also [10] Here Γ n = EK n is the edge set of the complete graph with n vertices and F n consists of the edge sets of copies of some fixed graph F say with k edges in K n Here the Erdős-Stone theorem implies that every subgraph H K n which contains at least χf 1 + o1 n 2 edges must contain a copy of F, where χf denotes the chromatic number of F see, eg, [2, 3, 5, 7] We are interested in random versions of such extremal results We study the binomial model of random substructures For a finite set Γ n and a probability p [0, 1] we denote by Γ n,p the random subset where every x Γ n is included in Γ n,p independently with probability p In other words, Γ n,p is the finite probability space on the power set of Γ n in which every elementary event {G} for G Γ n Date: October 23, Mathematics Subject Classification 05C80 11B25, 05C35 Key words and phrases Szemerédi s theorem, Turán s theorem, Erdős-Stone theorem, random sets, thresholds 1

3 2 MATHIAS SCHACHT occurs with probability P G = Γ n,p = p G 1 p Γn G For example, if Γ n is the edge set of the complete graph on n vertices, then Γ n,p denotes the usual binomial random graph Gn, p see, eg, [4, 25] The deterministic extremal results mentioned earlier can be viewed as statements which hold with probability 1 for p = 1 and it is natural to investigate the asymptotic of the smallest probabilities for which those results hold In the context of Szemerédi s theorem for every k 3 and ε > 0 we are interested in the smallest sequence p = p n n N of probabilities such that the binomial random subset [n] pn has asymptotically almost surely aas, ie with probability tending to 1 as n the following property: Every subset Y [n] pn with Y ε [n] pn contains an arithmetic progression of length k Similarly, in the context of the Erdős-Stone theorem, for every graph F and ε > 0 we are interested in the asymptotic of the smallest sequence p = p n n N such that the random graph Gn, p n aas satisfies: every H Gn, p with eh 1 1 χf 1 + ε egn, p n, contains a copy of F We determine the asymptotic growth of the smallest such sequence p of probabilities for those and some related extremal properties including multidimensional versions of Szemerédi s theorem Theorem 23, solutions of density regular systems of equations Theorem 24, an extremal version for solutions of the Schur equation Theorem 25, and extremal problems for hypergraphs Theorem 27 In other words, we determine the threshold for those properties Similar results were obtained by Conlon and Gowers [6] The new results will follow from a general result see Theorem 33, which allows us to transfer certain extremal results from the classical deterministic setting to the probabilistic setting In Section 4 we deduce the results stated in the next section from Theorem 33 2 New results 21 Szemerédi s theorem and its multidimensional extension We study extremal properties of random subsets of the first n positive integers One of the best known extremal-type results for the integers is Szemerédi s theorem In 1975 Szemerédi solved a longstanding conjecture of Erdős and Turán [13] by showing that every subset of the integers of upper positive density contains a arithmetic progression of any finite length For a set X [n] we write X ε [k] 1 for the statement that every subsets Y X with Y ε X contains an arithmetic progression of length k With this notation at hand, we can state the finite version of Szemerédi s theorem as follows: for every integer k 3 and ε > 0 there exists n 0 such that for every n n 0 we have [n] ε [k] For fixed k 3 and ε > 0 we are interested in the asymptotic behavior of the threshold sequence of probabilities p = p n such that there exist constants

4 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 3 0 < c < C for which lim P [n] q n ε [k] = n { 0, if q n cp n for all n N, 1, if q n Cp n for all n N 2 Remark 21 We note that the family {X [n]: X ε [k]} is not closed under supersets In other words, the property is X ε [k] is not a monotone property However, similar arguments as presented in [25, Proposition 86] show that the property X ε [k] and the other properties considered in this section have a threshold as displayed in 2 It is easy to see that if the expected number of arithmetic progressions of length k in [n] qn is asymptotically smaller than the expected number of elements in [n] qn, then there exists a subset of size 1 o1 [n] qn, which contains no arithmetic progressions of length k at all In other word, if q k nn 2 q n n q n n 1/k 1 3 then P [n] qn ε [k] 0 for every ε < 1 Consequently, n 1/k 1 is a lower bound on the threshold for Szemerédi s theorem for arithmetic progressions of length k For k = 3 Kohayakawa, Luczak, and Rödl [28] established a matching upper bound Our first result generalizes this for arbitrary k 3 Theorem 22 For every integer k 3 and every ε 0, 1 there exit constants C > c > 0 such that for any sequence of probabilities q = q n n N we have { lim P [n] 0, if q n cn 1/k 1 for all n N, q n ε [k] = n 1, if q n Cn 1/k 1 for all n N We remark that the 0-statement in Theorem 22 and, similarly, the 0-statements of the other results of this section follows from standard probabilistic arguments The 1-statement of Theorem 22 follows from our main result, Theorem 33 A multidimensional version of Szeméredi s theorem was obtained by Furstenberg and Katznelson [18] Those authors showed that for every integer l, every finite subset F N l and every ε > 0 there exists some integer n 0 such that for n n 0 every Y [n] l with Y εn l contains a homothetic copy of F, ie, there exist some y 0 N l and λ 0 such that y 0 + λf = {y 0 + λf : f F } Y Clearly, the case l = 1 and F = [k] resembles Szemerédi s theorem Generalizing the notation introduced in 1, for sets X, F N l and ε > 0 we write X ε F, if every subset Y X with Y ε X contains a homothetic copy of F A simple heuristic, similar to the one in the context of Szeméredi s theorem, suggests that n 1/ F 1 is a lower bound on the threshold for the Furstenberg- Katznelson theorem for a configuration F N l in the binomial random subset [n] l p where elements of [n] l are included with probability p Our next result shows that, in fact, this gives the correct asymptotic for the threshold Theorem 23 For every integer l 1, every configuration F N l with F 3, and every ε 0, 1 there exit constants C > c > 0 such that for any sequence of probabilities q = q n n N we have lim P [n] l q n n ε F = { 0, if q n cn 1/ F 1 for all n N, 1, if q n Cn 1/ F 1 for all n N

5 4 MATHIAS SCHACHT 22 Density regular matrices Another extension of Szemerédi s theorem leads to the notion of density regular matrices Arithmetic progressions of length k can be viewed as the set of distinct-valued solutions of the following homogeneous system of k 2 linear equations x 1 2x 2 + x 3 = 0, x 2 2x 3 + x 4 = 0, x k 2 2x k 1 + x k = 0 More generally, for an l k integer matrix A let SA R k be the set of solutions of the homogeneous system of linear equations given by A Let S 0 A SA be those solutions x 1,, x k with all x i being distinct We say A is irredundant if S 0 A Moreover, an irredundant l k integer matrix A is density regular, if for every ε > 0 there exists an n 0 such that for all n n 0 and every Y [n] with Y εn we have Y k S 0 A Szemerédi s theorem, for example, implies that the following k 2 k matrix is density regular for any k 3 Density regular matrices are a subclass of so-called partition regular matrices This class was studied and characterized by Rado [34] and, for example, it follows from this characterization that k l + 2 see [22] for details In [14] Frankl, Graham, and Rödl characterized irredundant, density regular matrices, being those partition regular matrices A for which 1, 1,, 1 SA Similar as in the context of Theorem 22 and Theorem 23 the following notation will be useful For an irredundant, density regular, l k integer matrix A, ε > 0, and X [n] we write X ε A if for every Y X with Y ε X we have Y k S 0 A The following parameter in connection with Ramsey properties of random subsets of the integers with respect to irredundant, partition regular matrices was introduced by Rödl and Ruciński [36] Let A be an l k integer matrix and let the columns be indexed by [k] For a partition W W [k] of the columns of A, we denote by A W the matrix obtained from A by restricting to the columns indexed by W Let ranka W be the rank of A W, where ranka W = 0 for W = We set ma = max W W =[k] W 2 W 1 W 1 + ranka W ranka 5 It was shown in [36, Proposition 22 ii ] that for irredundant, partition regular matrices A the denominator of 5 is always at least 1 For example, for A given in 4 we have ma = k 1 It follows from the 0-statement of Theorem 11 in [36] that for any irredundant, density regular, l k integer matrix A of rank l and every 1/2 > ε > 0 there exist a c > 0 such that for every sequence of probabilities q = q n with q n cn 1/mA

6 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 5 we have lim P [n] q n ε A = 0 6 n We deduce a corresponding upper bound from Theorem 33 and obtain the following result Theorem 24 For every irredundant, density regular, l k integer matrix A with rank l, and every ε 0, 1/2 there exit constants C > c > 0 such that for any sequence of probabilities q = q n n N we have lim P [n] q n ε A = n { 0, if q n cn 1/mA for all n N, 1, if q n Cn 1/mA for all n N Note that we restrict ε < 1/2 here With this restriction the 0-statement will follow from a result of Rödl and Ruciński from [36] The proof of the 1-statement presented in Section 42 actually works for all ε 0, 1 23 An extremal problem related to Schur s equation In 1916 Schur [38] showed that every partition of the positive integers into finitely many classes contains a class which contains a solution of the single, homogeneous equation x 1 + x 2 x 3 = 0 Clearly, the corresponding matrix is not density regular, since the set of all odd integers contains no solution However, it is not hard to show that that every subset Y [n] with Y 1/2 + o1n contains such a solution Similarly, as above for ε > 0 and X [n] we write X 1/2+ε if every subset Y X with Y 1/2 + ε X contains a distinct-valued solution, ie, Y k S We are interested in the threshold for the extremal problem of Schur s equation, ie, for the property X 1/2+ε In this context the simple heuristic based on the expected number of solutions of the Schur equation in random subsets of the integers suggests that n 1/2 is the threshold for this property Moreover, for Schur s theorem in random subsets of the integers the threshold turned out to be n 1/2 as shown in [21, 16] We show that the threshold of the extremal version of Schur s equation is the same Theorem 25 For every ε 0, 1/2 there exit constants C > c > 0 such that for any sequence of probabilities q = q n n N we have lim P { 0, if q n cn 1/2 for all n N, [n] qn 1/2+ε = n 1, if q n Cn 1/2 for all n N 24 Extremal problems for hypergraphs The last result deals with extremal problems for hypergraphs An l-uniform hypergraph H is a pair V, E, where the vertex set V is some finite set and the edge set E V l is a subfamily of the l-element subsets of V As usual we call 2-uniform hypergraphs simply graphs For some hypergraph H we denote by V H and EH its vertex set and its edge set and we denote by vh and eh the cardinalities of those sets For an integer n we denote by K n l the complete l-uniform hypergraph on n vertices, ie, vk n l = n and ek n l = n l An l-uniform hypergraph H is a sub-hypergraph of H, if V H V H and EH EH and we write H H to denote that For a subset U V H we denote by EU the edges of H contained in U and we set

7 6 MATHIAS SCHACHT eu = EU Moreover, we write H[U] for the sub-hypergraph induced on U, ie, H[U] = U, EU For two l-uniform hypergraphs F and H we say H contains a copy of F, if there exists an injective map ϕ: V F V H such that ϕe EH for every e EF If H contains no copy of F, then we say H is F -free We denote by exh, F the maximum number of edges of an F -free sub-hypergraph of H, ie, exh, F = max{eh : H H and H is F -free} Mantel [33], Erdős [8], and Turán [41] were the first to study this function for graphs In particular, Turán determined exk n, K k for all integers n and k This line of research was continued by Erdős and Stone [12] and Erdős and Simonovits [10] and those authors showed that for every graph F with chromatic number χf 3 we have exk n, F = 1 1 χf 1 + o1 n, 7 2 where χf is minimum number r such that there exists a partition V 1 V r = V F such that EV i = for every i [r] Moreover, it follows from the result of Kövari, Sós, and Turán [31] that exk n, F = on 2 8 for graphs F with χf 2 For an l-uniform hypergraph F we define the Turán density πf = lim n exk n l, F n l For a graph F the Turán density πf is determined due to 7 and 8 For hypergraphs 8 was extended by Erdős [9] to l-partite, l-uniform hypergraphs Here an l-uniform hypergraph F is l-partite if its vertex set can be partitioned into l classes, such that every edge intersects every partition class in precisely one vertex Erdős showed that πf = 0 for every l-partite, l-uniform hypergraph F For other l-uniform hypergraphs only a few results are known and, for example, determining πk 3 4 is one of the best known open problems in the area However, one can show that πf indeed exists for every hypergraph F see, eg [26] We study the random variable exg l n, q, F for l-uniform hypergraphs F, where G l n, q denotes the binomial random l-uniform sub-hypergraph of K n l with edges of K n l included independently with probability q It is easy to show that exh, F πf eh for all l-uniform hypergraphs H and F see, eg [25, Proposition 84] for a proof for graphs We are interested in the threshold for the property that aas exg l n, q, F πf + o1eg l n, q 9 Results of that sort appeared in the work of Babai, Simonovits, and Spencer [1] who showed that 9 holds for F = K 3 and q = 1/2 In fact, it follows from an earlier result of Frankl and Rödl [15] that the same holds as long as q n 1/2 The systematic study for graphs was initiated by Kohayakawa and his coauthors In particular, Kohayakawa, Luczak, and Rödl formulated a conjecture for the threshold of Turán properties for random graphs see Conjecture 26 below

8 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 7 For an l-uniform hypergraph F with ef 1 we set mf = max df with df = F F ef 1 { ef 1 vf l, if vf > l 1/l, if vf = l If follows from the definition of mf, that if q = Ωn 1/mF then aas the number of copies of every sub-hypergraph F F in the random hypergraph G l n, q has at least the same order of magnitude, as the number of edges Recall that a similar heuristic gave rise to the thresholds in the theorem above Conjecture 26 [29, Conjecture 1 i ] For every graph F with at least one edge and every ε > 0 there exists C > 0 such that for every sequence of probabilities q = q n n N with q n Cn 1/mF we have lim P exgn, q n, F πf + εegn, q n = 1 n Conjecture 26 was verified for a few special cases As already mentioned for F = K 3 the conjecture follow from a result in [15] For F being a clique with 4, 5, or 6 vertices the conjecture was verified by Kohayakawa, Luczak, and Rödl [29], Gerke, Schickinger, and Steger [20] and Gerke [19] Moreover, the conjecture is known to be true when F is a cycle due to the work of Füredi [17] for the cycle of length four and Haxell, Kohayakawa, and Luczak [23, 24] see also [27, 32] and the conjecture is known to be true for trees The best current bounds on q for which 9 holds for F being a clique and for arbitrary F were obtained by Szabó and Vu [39] and Kohayakawa, Rödl, and Schacht [30] We verify this conjecture for all graphs F and the natural analogue of this conjecture for hypergraphs For l-partite, l-uniform hypergraphs such a conjecture was made in [37, Conjecture 15] Theorem 27 For every l-uniform hypergraph F with at least one vertex contained in at least two edges and every ε 0, 1 πf there exist constants C > c > 0 such that for any sequence of probabilities q = q n n N we have lim P n ex G l n, q n πf + εe G l n, q n = { 0, if q n cn 1/mF for all n N, 1, if q n Cn 1/mF for all n N In Section 4 we will deduce the 1-statements of Theorems 23, 24, 25, and 27 from the main result, Theorem 33, which we present in the next section The proofs of the 0-statements will be more elementary and will be also given in Section 4 3 Main technical result The main result will be phrased in the language of hypergraphs We will study sequences of hypergraphs H = H n = V n, E n n N In the context of Theorem 22 one may think of V n = [n] and E n being the arithmetic progressions of length k In the context of Theorems 23, 24, and 25 the corresponding hypergraphs the reader should have in mind are defined in a very similar way Moreover, for Theorem 27 one should think of V n = EK l n hypergraph K l n 10 being the edge set of the complete and edges of E n correspond to copies of F in K l n

9 8 MATHIAS SCHACHT In order to transfer an extremal result from the classical, deterministic setting to the probabilistic setting we will require that a stronger quantitative version of the extremal result holds see Definition 31 below Roughly speaking, we will require that a sufficiently dense sub-structure not only contains one copy of the special configuration not only one arithmetic progression or not only one copy of F, but instead the number of those configurations should be of the same order as the total number of those configurations in the given underlying ground set Definition 31 Let H = H n n N be a sequence of k-uniform hypergraphs and α 0 We say H is α-dense if the following is true For every ε > 0 there exist ζ > 0 and n 0 such that for every n n 0 and every U V H n with U α + ε V H n we have EH n [U] ζ EH n The second condition in Theorem 33 imposes a lower bound on the smallest probability for which we can transfer the extremal result to the probabilistic setting see Definition 32 For a k-uniform hypergraph H = V, E, i [k 1], v V, and U V we denote by deg i v, U the number of edges of H containing v and having at least i vertices in U \ {v} More precisely, deg i v, U = {e E : e U \ {v} i and v e} 11 For q 0, 1 we let µ i H, q denote the expected value of the sum over all such degrees squared with U = V q being the binomial random subset of V [ ] µ i H, q = E deg 2 i v, V q v V Definition 32 Let H = H n n N be a sequence of k-uniform hypergraphs, let p = p n n N 0, 1 N be a sequence of probabilities, and let K 1 We say H is K, p-bounded if the following is true For every i [k 1] there exists n 0 such that for every n n 0 and q p n we have µ i H n, q Kq 2i EH n 2 V H n 12 With those definitions at hand, we can state the main result Theorem 33 Let H = H n = V n, E n n N be a sequence of k-uniform hypergraphs, let p = p n n N 0, 1 N be a sequence of probabilities satisfying p k n E n as n, and let α 0 and K 1 If H is α-dense and K, p-bounded, then the following holds For every δ > 0 and ω n n N with ω n as n there exists C 1 such that for every q n Cp n the following holds aas for V n,qn : If W V n,qn with W α + δ V n,qn, then EH n [W ] The proof of Theorem 33 is based on induction on k and for the induction we will strengthen the statement see Lemma 34 below For a k-uniform hypergraph H = V, E subsets W U V, and any integer i {0, 1,, k} we consider those edges of H[U] which have at least i vertices in W and we denote this family by E i U W = {e EH[U]: e W i}

10 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 9 Note that EU 0 W = EH[U] and EU k W = EH[W ] 13 for every W U Lemma 34 Let H = H n = V n, E n n N be a sequence of k-uniform hypergraphs, let p = p n n N 0, 1 N be a sequence of probabilities satisfying p k n E n as n, and let α 0 and K 1 If H is α-dense and K, p-bounded, then the following holds For every i [k], δ > 0, and ω n n N with ω n as n there exist ξ > 0, b > 0, C 1, and n 0 such that for all β, γ 0, 1] with βγ α + δ, every n n 0, every q with 1/ω n q Cp n the following holds If U V n with U β V n, then the binomial random subset U q satisfies with probability at least 1 2 bq Vn the following property: For every subset W U q with W γ U q we have E i U W ξq i E n Theorem 33 follows directly from Lemma 34 applied with i = k, β = 1, γ = α + δ, and U = V n 31 Probabilistic tools We will use Chernoff s inequality in the following form see, eg, [25, Corollary 23] Theorem 35 Chernoff s inequality Let X Y be finite sets and p 0, 1] For every 0 < ϱ 3/2 we have P X Yp p X ϱp X 2 exp ϱ 2 p X /3 We also use an approximate concentration result for K, p-bounded hypergraphs The K, p-boundedness only bounds the expected value of the quantity v deg2 i v, V p In the proof of Lemma 34 we need an exponential upper tail bound and, unfortunately, it is known that such bounds usually not exist However, it was shown by Rödl and Ruciński in [35] that on the prize of deleting a few elements such bound can be obtained We will again apply this idea in the proof of Lemma 34 Proposition 36 Upper tail [35, Lemma 4] Let H = H n = V n, E n n N be a sequence of k-uniform hypergraphs, let p = p n n N 0, 1 N be a sequence of probabilities, and let K 1 If H is K, p-bounded, then the following holds For every i [k 1] and every η > 0 there exist b > 0 and n 0 such that for every n n 0 and every q p n the binomial random subset V n,q has the following property with probability at least 1 2 bq Vn +1+log 2 k There exists a set X V n,q with X ηq V n such that deg 2 i v, V n,q \ X 4 k k 2 2i En 2 Kq V n v V n The proof follows the lines of [35, Lemma 4] and we include it for completeness Proof Suppose H is K, p-bounded and i [k 1] and η > 0 are given We set η b = 4k 1 2 and n 0 be sufficiently large, so that 12 holds for every n n 0 and q p n

11 10 MATHIAS SCHACHT For every j = i,, 2k 1 we consider the family S j defined as follows { S j = S, v, e, e : S V n, v V n, e, e E n such that S = j, } v e e, S e e \ {v}, e S i and e S i Let S j be the random variable denoting the number of elements S, v, e, e from S j with S V n,q 2k 2 j By definition we have j=i E [S j ] 4 k 1 µ i H n, q and due to the K, p-boundedness of H we have 2k 2 max E [S j] E [S j ] 4 k 1 µ i H n, q 4 k 1 Kq 2i E n 2 j=i,,2k 1 V n j=i Let Z j be the random variable denoting the number of sequences of length which satisfy z = S r, v r, e r, e r r [z] S z j ηq Vn ηq Vn 4k 1 2 2k 1j i the sets S r are contained in V n,q and ii the sets S r are mutually disjoint, ie, S r1 S r2 = for all 1 r 1 < r 2 z Clearly, we have E [Z j ] S j z q jz = E [S j ] z 4 k 1 Kq 2i E n 2 z V n On the other hand, if v V n deg 2 i v, V n,q \ X 4 k k 2 Kq 2i E n 2 V n 2k 2 j=i j 2 4 k 1 Kq 2i E n 2 V n for any X V n,q with X ηq V n, then there exists some j 0 {i,, 2k 2} such that Z j0 2 4 k 1 Kq 2i E n 2 z V n Markov s inequality bounds the probability of this event by P j 0 {i,, 2k 2}: Z j0 2 4 z k 1 Kq 2i E n 2 z V n 2k 2 P Z j 2 4 z k 1 Kq 2i E n 2 z 2k 2 z 2 bq Vn +1+log 2 k, V n j=i which concludes the proof of Proposition 36

12 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES Proof of Lemma 34 Let H = H n = V n, E n n N be a sequence of k- uniform hypergraphs, let p = p n n N 0, 1 N be a sequence of probabilities, and let α 0 and K 1 such that H is α-dense and K, p-bounded We prove Lemma 34 by induction on i Induction start i = 1 For δ > 0 and ω n n N which plays no role for the induction start we appeal to the α-denseness of H and let ζ and n 1 be the constants given by this property for ε = δ/8 We set ξ = δζ 4k, b = δ3 193, C = 1, and n 0 = n 1 Let β, γ 0, 1] satisfy βγ α + δ, let n n 0 be sufficiently large, q p n, and let U V n with U β V n be given We consider the set Y U defined by { Y = u U : {e EH n [U]: u e} ζ E } n 2 V n In other words, Y is the set of vertices in U with low degree in H n [U] Due to the α-denseness of H we have Y α + δ V n 8 It follows from Chernoff s inequality that with probability at least 1 2 exp δ 2 q U /48 2 exp δ 2 q V n / bq Vn we have U q 1 δ q U and U q Y α + δ q V n 4 4 Consequently, for every W U q satisfying W γ U q we have W γ U q 1 δ γq U 1 δ βγq V n δ α + δ q V n α + δ q V n U q Y + δ q V n and the definition of Y yields E 1 U W 1 ζ E n W \ Y k 2 V n δ 2 q V n 1 ζ E n k 2 V n = ξq E n This concludes the proof of the induction start Induction step i i + 1 Let i 1, δ > 0, and ω n n N with ω n as n be given We will expose the random set U q in several rounds The number of main rounds R will depend on the constant ξi, δ/8, which is given by the induction assumption More precisely, let ξ = ξi, δ/8, b = bi, δ/8, C = Ci, δ/8, and n = n 0 i, δ/8 be given by the induction assumption applied with δ = δ/8 We set 4 k+2 k 2 K R = δξ Overview Roughly, speaking our argument is as follows We will expose U q in R main rounds of the same weight, ie, we will chose q R in such a way that 1 q = 1 q R R and we let U q = U 1 q R U R q R Since, every subset W,

13 12 MATHIAS SCHACHT which we have to consider, contains at least γ α + δ proportion of the elements of U q there must be at least δr/4 rounds such that Uq s R W α + δ/2 U qr For those rounds we will appeal to the induction assumption, which combined with Proposition 36, implies that U contains at least Ωξ 2 V n elements u U with the property that every such u completes many elements in EU i W U q s R to elements in E i+1 U W U q s R Moreover, in each of these substantial rounds ξ 2 V n /4 k+1 k 2 K new rich elements u will be created Consequently, after at most δr/4 1 of these substantial rounds all but, say, at most α+δ/8 V n < γ V n elements of U are rich and in the final substantial round W U qr must contain many rich u Uand therefore create many elements from E i+1 U W However, the error probabilities in the later rounds will have to beat the number of choices for the elements of W in the earlier rounds For that we will split the earlier main rounds into several subrounds This does not affect the argument indicated above, since our bound on the number of rich elements will be independent of q R We now continue with the details of this proof Constants Set η = δ2 9 and let ˆb and ˆn be given by Proposition 36 applied with i and η We set Finally, let b = min { δ , b 3, ˆb 3 } 15 and B = ξ δ 2 ξ = 18kRB R 1, 17 i+1 { δ 3 b = min 60001RB R 1, b } 2RB R 1, 18 C = RB R 1 C, 19 and let n 0 max{n, ˆn} be sufficiently large Suppose β and γ 0, 1] satisfy βγ α + δ Let n n 0 and let q satisfy 1/ω n q Cp n Moreover, let U V n be such that U β V n Note that min{β, γ} α + δ δ > 0 and U α + δ V n For a simpler notation from now on we suppress the subscript n in p n, H n, V n and E n Details of the induction step As discussed above we generate the random set U q in several rounds We will have R main rounds and for that we choose q R such that 1 q = 1 q R R For s [R] we will further split the sth main round into B R s subrounds For s [R] we set r s = B R s b

14 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 13 and let q s satisfy 1 q R = 1 q s rs Note that for sufficiently large n, due to q n 1/ω n and ω n we have 1 + δ q 100 R q R q R and 1 + δ qr 100 B R s q s q R, 20 BR s and due to the choice of B we have s 1 t=1 q t 101 q s 1 R B R t=1 B t 16 b q R 101 B R Bs b 101 q s 21 We proceed as follows we first consider r 1 rounds with probability q 1, which all together establish the first main round and we denote the random subsets obtained by U 1 q R = U 1,1 q 1 U 1,r1 q 1 This is followed by r 2 rounds with probability q 2 establishing the second main round This way we have and for all s [R] U q = U 1 q R U R q R U s q R = U s,1 q s U 1,rs q s Furthermore, let W G q with W γ U q and let W s = W U s q R and W s,j = W U s,j q s for all s [R] and j [r s ] In our analysis we focus on substantial rounds For that let S [R] be the set defined by s S if and only if W s γ δ Uq s 2 R By definition of S, for every s S exists some j s [r s ] such that W s,js γ δ Uq s,js 2 s and for the rest of the proof we fix such an j s for every s S The following claim is a direct consequence of Chernoff s inequality Claim 1 Let A denote the event that S δr/4 Then P A 1 2 2bq V Proof Due to Chernoff s inequality we have U s,j q s = 1 ± 001δq s U 22 for all s [R] and every j [r s ] with probability at least 1 2 R r s exp δ 2 q s U / bq V, s=1

15 14 MATHIAS SCHACHT where we used q 1 q s, 20, the choice of b in 18 and the fact that n is sufficiently large for the last inequality Since W γ U q we have S W R 1 + δ/100γ δ/2q R U 1 + δ/100q R U 1 δ/100γq γ δ R 20 δ 1 + δ/100q R 2 4 R with probability at least 1 2 2bq V For the rest of the proof we analyze the rounds indexed by s, j s for s S For s S we set W s = W t,jt and Us = Uq t,jt t t S t S t s t s Note that W t = Ut = for all t < min s S s Roughly, speaking we will show for every s S that either E i+1 U W s is sufficiently large or Ω V new rich elements in U will be created More precisely, for s S we consider the following subset Z s U of rich elements } Z s := {u U : deg i u, W s,js, U ξ E 2 qi s, V where deg i u, W s,js, U := { e E : e W s,js \ {u} i and e U } 23 Note that deg i u, W s,js, V = deg i u, W s,js and, hence, for every set U V and every u V we have deg i u, W s,js, U deg i u, W s,js 24 Similarly, as above we set Zs = t S t s Z s Claim 2 For every s S and any choice of W s 1 Us 1 let B W s 1 denote the event that Uq s,js s satisfies the following properties: Then i Uq s,js s ii for every W s,js or 101q s U and with W s,js γ δ/2 U s,js q s either E i+1 U W s ξqi+1 E 25 Zs \ Zs 1 ξ 2 4 k+1 k 2 V 26 K P B W s 1 Us b q s V, where P B W s0 1 Us 0 1 = P B W s0 1 for s0 = min s S s

16 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 15 Before we verify Claim 2 we deduce Lemma 34 from it Let C denote the event that the conclusion of Lemma 34 holds If event A holds and B W s 1 holds for every s S, then C must hold, since 26 in Claim 2 can occur at most 4 k+1 k 2 K ξ 2 14 < δ 4 R S times and, therefore, 25 in Claim 2 must occur Below we will verify that this happens with a sufficiently large probability Setting P Us 0 1 = 1 for s 0 = min s S s, we have P C P A + P B W s 1 Us 1 P Us 1, S [R] s S Us 1 W s 1 where the first sum runs over all subsets S [R] with S δr/4, the third sum runs over all choices of Us 1 = t S,t<s U q t,jt t with Uq t,jt t 101q t G, and the inner sum runs over all V t S,t<s qt choices of W s 1 Us 1 Therefore, Claims 1 and 2 yield P C 2 2bq V + 2 R R s= V s 1 t=1 qt 2 2b q s V bq V + 2 R R2 b q 1 V bq V + 2 R R2 b q V /RB R bq V, where the last inequality holds for sufficiently large n This concludes the proof of Lemma 34 and it is left to verify Claim 2 Proof of Claim 2 Let s S, W s 1 Us 1 be given Note that this also defines Zs 1 We first observe that property i of Claim 2 holds with high probability In fact, due to Chernoff s inequality, with probability at least 1 2 exp δ 2 q s U / b q s V we even have Uq s,js s = 1 ± 001δq s U 27 and below we assume that 27 holds We distinguish two cases for property ii Case 1 U \ Zs 1 < γ 3δ/4 U Due to Chernoff s inequality with probability at least we have Uq s,js s Since s S it follows that 1 2 exp δ 2 α + δ/4q s U / b q s V \ Zs 1 γ 5 8 δ U s,js q s Wq s,js s Zs 1 δ s,js Uq 8 s δ 8 q s U δβ 9 q s V δ2 9 q s V Hence the definition of Zs 1 t S,t<s Zs and q 1 q t for all t S yields E i+1 U δ2 W s 9 q s V 1 k ξ 2 qi 1 E V ξ δ 2 18k qi+1 1 E 20 ξ δ 2 18kRB R 1 i+1 qi+1 E 17 ξq i+1 E

17 16 MATHIAS SCHACHT In other words, for this case we showed that alternative 25 happens with probability at least b q s V 1 2 2b q s V Case 2 U \ Zs 1 γ 3δ/4 U In this case we consider We set Clearly, β 0, 1], and β γ = β = U V U = U \ Zs 1 and γ = γ 7δ U 8 U 0 < γ γ 7δ/8 γ 3δ/4 1, γ 7δ U 8 V γ 7δ β γβ 7δ 8 8 α + δ 8 Hence, we can apply the induction assumption to U More precisely, the induction assumption asserts that with probability at least 1 2 b q s V every subset Ŵ U q s with Ŵ γ U q s satisfies EU i Ŵ ξ qs E i 28 Note that, in fact, q s 20 q RB R 1 Cp 19 RB R 1 C p We split the random subset U s,js q s = U q s U q s, where U q s = Uq s,js s \ Zs 1 and U q s = Uq s,js s \ U q s Similarly, we split W s,js = W W where W = W s,js U q s and W = W s,js U q s It follows again from Chernoff s inequality that holds with probability at least U q s = 1 ± δ exp δ 2 q s U / b q s V We distinguish two sub-cases depending on the size of W q s U 29 Case 21 W > δ U s,js q s /8 In this case, it follows from the W Zs 1 E i+1 U W s W 1 ξ k 2 qi 1 27 δ 9 q s U ξ 2k qi 1 E V δ s,js Uq 8 s ξ 2k qi 1 E V E V δβ 9 q s ξ 2k qi 1 E δ2 ξ 18k qi+1 1 E 17 ξq i+1 E In other words, for this case we showed that alternative 25 happens with probability at least b q s V 1 2 2b q s V Case 22 W δ U s,js q s /8

18 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 17 In this case we appeal to the K, p-boundedness of H It follows from Proposition 36 and the choice of η in 15 that with probability at least 1 2 ˆbq s V +1+log 2 k there exists a set X U q s such that and X ηq s V 15 δ2 16 q s V δ 16 α + δq s V deg 2 i u, W \ X, U 24 deg 2 i u, W \ X u U u U δ 16 γq s V δ 16 q s U 27 δ 8 q s U s,js q s 30 deg 2 i u, U q s \ X 4 k k 2 Kq 2i E 2 s V 31 u U Consider the set Ŵ = W \ X Since s S, it follows from 30 and the assumption of this case that Ŵ W s,js W X γ δ Uq s,js 2 s 2 δ s,js Uq 8 s Furthermore assertions 27 and 29 yield Ŵ U q s γ 3δ U s,j s q s γ 3δ/41 δ/100 4 U q s 1 + δ/16 U U γ 3δ 4 U s,js q s γ 7δ U 8 U = γ In other words, Ŵ satisfies Ŵ γ U q s and from the induction assumption we infer that 28 holds with probability at least 1 2 b q s V and then u U deg i u, Ŵ, U E i U Ŵ ξ q i s E 32 For Ẑ = {u U : deg i u, Ŵ, U ξ it follows from the Cauchy-Schwarz inequality 2 qi s } E V 4 k k 2 Kqs 2i E 2 31 deg 2 i u, V Ŵ, U deg 2 i u, Ŵ, U u U u Ẑ 1 Ẑ deg i u, Ŵ, U u Ẑ Ẑ ξ qs E i 2 2 Consequently, Ẑ ξ 2 4 k+1 k 2 K V

19 18 MATHIAS SCHACHT Since Ẑ U = U \ Zs 1 we have Ẑ is disjoint from Zs 1 Furthermore, by definition of Ẑ we have Ẑ Zs Therefore, 26 of Claim 2 holds with probability at least b q s V 2 ˆbq s V +1+log 2 k 2 b q s V b q s V, which concludes the proof of Claim 2 4 Proof of the new results In this section we prove Theorems 22, 23, 24, 25, and 27 While the involved 0-statements will follow from standard probabilistic arguments, the 1-statement of those results will follow from Theorem Proof of Theorems 22 and 23 Clearly Theorem 22 follows from Theorem 23 applied with l = 1 and F = [k] and it suffices to verify Theorem 23 The 0-statement of Theorem 23 We start with the 0-statement of the theorem Let F N l with F 3 and ε > 0 be given and set 1/ F 1 1 2ε c = 2 We distinguish different cases depending on the sequence q = q n Case 1 q n n l+1/ F In this case the expected number of homothetic copies of F in [n] l q n tends to 0 Hence, we infer from Markov s inequality that aas [n] l q n contains no homothetic copy of F, which yields the claim in that rage Case 2 n l q n n 1/ F 1 In this range the expected number of homothetic copies of F in [n] l q n is asymptotically smaller than the expected number of elements in [n] l q n Moreover, it follows from Chernoff s inequality that aas [n] l q n is very close to its expectation Consequently, it follows from Markov s inequality that aas the number of homothetic copies of F in [n] l q n is o [n] l q n Therefore, by removing one element from every homothetic copy of F in [n] l q n aas we obtain a subset Y of size Y ε [n] l q n, which contains no homothetic copy of F at all, which yields the 0-statement in this case Note that due to F 3 the ranges considered in Cases 1 and 2 overlap Similarly, the range considered in the case below overlaps with the one from Case 2 Case 3 n l+1/ F q n cn 1/ F 1 Again appealing to Chernoff s inequality applied to the size of [n] l q n we infer that it suffices to show that aas the number of homothetic copies of F in [n] l q n is at most 1 2εq n n l Let Z F E [Z F ] q n F be the random variable denoting the number of such copies Clearly, n l+1 and standard calculations show that the variance of Z F satisfies 2 F 1 Var [Z F ] = O qn n l+2 + q n F n l+1 Consequently, Chebyshev s inequality yields P Z F 2q n F n l+1 Var [Z F ] qn 2 F n = O 1 2l+2 q n n l + 1 = o1, q n F n l+1

20 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 19 due to the range of q n we consider in this case Hence, the claim follows from the choice of c, which yields 2q F n n l+1 1 2εq n n l The 1-statement of Theorem 23 We now turn to the 1-statement of Theorem 23 We first note that if q n = Ω1, then the theorem follows directly from Chernoff s inequality combined with the original result of Furstenberg and Katznelson Hence we can assume wlog q n = o1 Let F N l with k = F 3 and ε 0, 1 We want to apply Theorem 33 For that we consider the following sequence of k-uniform hypergraphs H = H n = V n, E n n N Let V n = [n] l and let every homothetic copy of F form an edge in E n In particular, E n = n l+1 We set p n = n 1/k 1, p = p n n N and α = 0 Clearly, for those definitions the conclusion of Theorem 33 yields the 1-statement of Theorem 23 In order to apply Theorem 33 we have to verify the following three conditions a p k n E n as n, b H is α-dense, and c H is K, p-bounded By definition of p n and H n we have p k n E n = Ω n k/k 1 n l+1 = Ω n l 1/k 1, which yields a, as l 1 and k 3 Condition b holds, due to work of Furstenberg and Katznelson [18] In fact, it follows from the result in [18], that for every configuration F N l and every ε > 0 there exist ζ > 0 and n 0 such that for every n n 0 every subset U [n] l with U εn l contains at least ζn l+1 homothetic copies of F In other words, H is 0-dense Hence, it is only left to verify condition c We have to show that for every i [k 1] and q p n = n 1/k 1 we have [ ] µ i H n, q = E deg 2 i v, V q = O q 2i E n 2 = O q 2i n l+2 33 V n v V n It follows from the definition of deg i in 11 that µ i H n, q is the expected number of pairs F 1, F 2 of homothetic copies of F which share at least one point v and at least i points different from v of each copy are contained in [n] l q The expected number of such pairs F 1, F 2 which share exactly one point can be bounded by O q 2i n l+2 Since for every fixed homothetic copy F 1 there exist only constantly many independent of n other copies F 2, which share two points with F 1, the expected number of such pairs F 1, F 2 with F 1 F 2 2 is bounded by O q i n l+1 = O q 2i n l+2, since q Cp n Cn 1/k 1 Cn 1/i Consequently, 33 holds, which concludes the proof of Theorem Proof of Theorems 24 The proof of the 0-statement follows directly from the 0-statement of Theorem 11 in [36] Those authors showed that for every irredundant, density regular l k matrix with rank l there exists a constant c > 0 such that for q n cn ma aas [n] qn can be partitioned into two classes such that

21 20 MATHIAS SCHACHT none of them contains a distinct-valued solution of the homogeneous system given by A Clearly, this implies the 0-statement of Theorems 24 for every ε 0, 1/2 The 1-statement of Theorem 24 First we note that if q n = Ω1, then the statement follows directly from Chernoff s inequality combined with the definition of irredundant, density regular matrix Let A be an irredundant, density regular l k integer matrix and ε > 0 For the application of Theorem 33 we consider the following sequence of k-uniform hypergraphs H = H n = V n, E n n N Let V n = [n] and for every distinctvalued solution x 1,, x k let {x 1,, x k } be an edge of E n Moreover we set p n = n 1/mA, p = p n n N and α = 0 The 1-statement of Theorem 24 then follows from the conclusion of Theorem 33 and we have to verify the same three conditions a -c as in the proof of the 1-statement of Theorem 23 It was shown in [36, Proposition 22 ii ] that ma k 1 and due to Rado s characterization of partition regular matrices which contains the class of all density regular matrices we have k l 2, which yields E n = Ωn 2 Therefore, we have p k E n = Ωn k/k 1 n 2 = Ω n k 2 k 1 and, hence, condition a is satisfied Moreover, based on the Furstenberg-Katznelson theorem from FuKa78 it was shown by Frankl, Graham, and Rödl in [14, Theorem 2], that the sequence of hypergraphs H defined above is 0-dense, ie, condition b is fulfilled Consequently, it suffices to verify that H is K, p-bounded for some K 1 For i [k 1] and q p n = n 1/mA we have to show that µ i H n, q = O q 2i E n 2 n Recalling the definitions of µ i H n, q and H n = [n], E n we have µ i H n, q = E deg 2 i x, V n,q = E [ deg 2 i x, V n,q ] 34 x [n] x [n] Note that E [ deg 2 i x, V n,q ] is the expected number of pairs X, Y [n] k [n] k such that i x X Y, ii X = {x 1,, x k } and Y = {y 1,, y k } are solutions of LA, where Ax = Ay = 0 for x = x 1,, x k t and y = y 1,, y k t, and iii X [n] q \ {x} i and Y [n] q \ {x} i For fixed x and X, Y let w 1 be the largest integer such that there exist indices i 1,, i w and j 1,, j w for which Consequently, x i1 = y j1,, x iw = y jw 35 x {x i1,, x iw } = {y j1,, y jw } 36 Set W 1 = {i 1,, i w } and W 2 = {j 1,, j w } For fixed sets W 1, W 2 [k] we are going to describe all 2k w-tuples X Y satisfying ii and 35 To this end consider the 2l 2k w matrix B, which arises

22 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 21 from two copies A 1 and A 2 of A with permuted columns We set A 1 = A W 1 A W1 and A 2 = A W2 A W 2 where for every α = 1,, w the column of A W1 which is indexed by i α aligns with that column of A W2 which is indexed by j α Then let AW 1 A W1 0 B = 0 A W2 A W 2 Without loss of generality we may assume that ranka W 1 ranka W 2 and, therefore, rankb ranka + ranka W 1 Clearly, the number of 2k w-tuples X Y satisfying ii and 35 equals the number of solutions of the homogeneous system given by B, which is On 2k w rankb Since A is an irredundant, partition regular matrix, it follows from [36, Proposition 22 i ] that ranka = ranka for every matrix A obtained from A by removing one column Consequently, any matrix B obtained from B by removing one of the middle columns ie, one of the w columns of B which consist of a column of A W1 and a columns of A W2 satisfies rankb ranka + ranka W 1 = l + ranka W 1 Therefore, it follows from 36 that the number of such 2k w-tuples that also satisfy condition i for some fixed x [n] is at most On 2k w 1 l ranka W 1 37 Finally, we estimate the probability that a 2k w-tuple X Y satisfying i, ii, and 35 also satisfies iii Since X Y [n] q \ {x} = j w 1 and q 1 this probability is bounded by In view of 37 we obtain E [ deg 2 i x, V n,q ] x [n] = w 1 k j=0 x [n] w=1 W 1,W 2 [k] W 1 = W 2 =w q 2i j = Oq 2i w+1 On 2k w 1 l ranka W 1 q 2i w+1 38 Note that if w = 1, then again due to [36, Proposition 22 i ] we have ranka W 1 = l and, therefore, the contribution of those terms satisfies On 2k 2l 2 q 2i = On 2k 2l 1 q 2i = O q 2i E n 2 39 n x [n] W 1,W 2 [k] W 1 = W 2 =1 For w 2 and W 1 [k] with W 1 = w we obtain from the definition of ma and q n 1/mA that q w 1 n w+1 ranka W 1 +l

23 22 MATHIAS SCHACHT Consequently, k x [n] w=2 W 1,W 2 [k] W 1 = W 2 =w On 2k w 1 l ranka W 1 q 2i w+1 = k x [n] w=2 W 1,W 2 [k] W 1 = W 2 =w On 2k 2 2l q 2i Finally, combining 34, 38, 39, and 40 we obtain µ i H n, q = O q 2i E n 2, n = On 2k 2l 1 q 2i = O q 2i E n 2 40 n which concludes the proof of the 1-statement of Theorem Proof of Theorems 25 The proof is similar to the proof of Theorem 23 and we only sketch the main ideas The 0-statement of Theorem 25 We recall that for the validity of the statement X 1/2+ε we only consider distinct-valued of the Schur equation and we call such a solutions Schur-triples The expected number of Schur-triples contained in [n] qn is bounded by qnn 3 2 Consequently, the 0-statement follows from Markov s inequality if q n n 2/3 In the middle range n 1 q n n 1/2 it follows, on the one hand, from Chernoff s inequality that aas [n] qn q n n/2 On the other hand, due to Markov s inequality aas the number of Schur-triples in [n] qn is oq n n and, hence, the statement holds in this range of q n Finally, if n 2/3 q n cn 1/2 for sufficiently small c > 0, then using Chebyshev s inequality one obtains the upper bound of 1 1/2 + εq n n/2 on the number of Schur-triples in [n] qn, which holds aas Consequently, in view of Chernoff s inequality, aas the random set [n] qn contains a subset of size 1/2 + ε [n] qn, which contains no Schur-triple The 1-statement of Theorem 25 Here the we consider a sequence of 3-uniform hypergraphs, where V n = [n] and E n corresponds to all Schur-triples in [n] and we set p n = n 1/2 and α = 1/2 For given ε 0, 1/2 we want to appeal to Theorem 33 and for that we assume q n = o1 Again the 1-statement of Theorem 25 follows from Theorem 33 and we have to verify the three conditions a -c as in the proof of the 1-statement of Theorem 23 Condition a follows from the definition of p n and condition c follows from similar considerations as in the proof of Theorem 23 for l = 1 and k = 3 In order to verify condition b we have to show that for every ε > 0 there exist ζ > 0 and n 0 such that for n n 0 every subset A [n] with A 1/2 + εn contains at least ζn 2 Schur-triples

24 EXTREMAL RESULTS FOR RANDOM DISCRETE STRUCTURES 23 So let A [n] satisfy 1/2 + εn and set A 1 = A {1,, 1 εn} ignoring floors and ceilings It follows that for every z A \ A 1 there are at least ε 1 εn n εn = ε 2 2 n pairs x y with x, y A such that x + y = z Hence, if A \ A 1 3ε 2 n/2, then A contains at least 3ε 3 n 2 /4 n Schur-triples and the claim follows On the other hand, if A \ A 1 < 3ε 2 n/2, then we have A ε n 3ε2 2 n = ε 2 1 εn In other words, we obtained a density increment of ε/2 on the interval 1 εn and the conclusion follows from iterating the above argument This concludes the proof of condition b and, therefore, Theorem 33 yields the proof of the 1-statement of Theorem 25 for sequences q satisfying q n = o1 The remaining case, when q n = Ω1 then follows by similar arguments as given in [25, Proposition 86] and we omit the details 44 Proof of Theorems 27 The 0-statement of Theorem 27 Let F be an l-uniform hypergraph with at least one vertex of degree 2 and ε 0, 1 πf We set c = 1 πf ε 4 For the proof of the 0-statement we consider different ranges of q = q n n N depending on the density of the densest sub-hypergraph of F and depending on mf Let F be the densest sub-hypergraph of F with ef 1, ie, F maximizes ef /vf Moreover, let F be one of those sub-hypergraphs for which df = mf see 10 for the definition of those parameters Note that ef 2, since F contains a vertex of degree at least two We consider the following three ranges for q Case 1 q n n vf /ef In this range the expected number of copies of F in G l n, q n tends to 0 and, therefore, the statement follows from Markov s inequality Case 2 n l q n n 1/mF It follows from the definition of mf, that in this range the expected number of copies of F in G l n, q n is asymptotically smaller than the expected number of of edges of G l n, q n Therefore, applying Markov s inequality to the number of copies of F and Chernoff s inequality to the number of edges G l n, q n we obtain that aas the number of copies of F satisfies oeg l n, q n Hence, aas we can obtain an F -free, and consequently, an F -free sub-hypergraph of G l n, q n by removing only oeg l n, q n edges, which yields the statement for this range of q n We note that n l n vf /ef since F contains a vertex of degree 2 In other words, the interval considered in Case 2 overlaps with the interval from Case 1 Similarly, the range considered in the case below overlaps with the one from Case 2

25 24 MATHIAS SCHACHT Case 3 n vf /ef q n cn 1/mF Applying again Chernoff s inequality to eg l n, q n we see that it suffices to show that aas the number of copies of F is at most 1 πf + εq n n l /2 Let Z F be the random variable denoting the number of such copies Clearly, E [Z F ] q ef n n vf and standard calculations show that the variance of Z F satisfies Var [Z F ] = O q 2eF n n 2vF min F F,eF 1 q ef n n vf = O q 2eF n n 2vF q ef n n vf due to the choice of F being the densest sub-hypergraph of F Since q n n vf /ef we have q ef n n vf and, therefore, Var [Z F ] = o q 2eF n n 2vF Consequently, Chebyshev s inequality yields P Z F 2q ef n n vf Var [Z F ] q 2eF n n = o1 2vF Moreover, since q n cn 1/mF and ef 2 it follows from the choice of c that, 2q ef n n vf which yields the 0-statement in this case 1 πf + ε q n n l, 2 The 1-statement of Theorem 27 Let F be an l-uniform hypergraph with at least Theorem 33 For that we consider the sequence of k-uniform hypergraphs H = H n = V n, E n n N where V n = EK n l and edges of E n correspond to copies of F in K n Moreover, we set p n = n 1/mF and α = πf Clearly, for this set up the conclusion of Theorem 33 yields the 1-statement of Theorem 27 for sequences q with q n = o1 In order to apply Theorem 33 we have to verify the three conditions a -c stated in the proof of the 1-statement of Theorem 23 Condition a follows from the definitions of p n and E n combined In fact, since F contains a vertex of degree at least 2 we have mf 1/l 1 and p n E n = Ωn Such a result was obtained by Erdős and Simonovits [11, Theorem 1] and, hence, it is left to verify condition c only To this end observe that H n is a regular hypergraph with n l vertices and every vertex is contained in Θn vf l edges and that E n = Θn vf We will show that for q n 1/mF and i [k 1] we have [ ] µ i H n, q = E deg 2 i v, V n,q = E [ deg 2 i v, V n,q ] = O q 2i E n 2 V n v V n v V Due to the definition of H every v V n corresponds to an edge ev in K n l Therefore, the number E [ deg 2 i v, V n,q ] is the expected number of pairs F 1, F 2 of copies F 1 and F 2 of F in K n l satisfying ev EF 1 EF 2 and both copies F 1 and F 2 have at least i edges in EG l n, q \ {ev} Summing over all such pairs