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A setAof positive integers is aBh-setif all the sums of the forma1+ . . . +ah, witha1,. . .,ah∈Aanda1⩽ . . . ⩽ah, are distinct. We provide asymptotic bounds for the number ofBh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context ofBh-sets. We also use these results to estimate the maximum size of aBh-sets contained in a typical (random) subset of [n] with a given cardinality.

A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\H{o}s characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to appear in a collection of a given number $a$ of subsets of an $n$-element set. For every $a$, this minimum is achieved by the collection comprising $a$ sets whose cardinalities are as close to $n/2+1/4$ as possible. We show that the same is true about chains of an arbitrary length $k$, for all $a$ and $n$, confirming the prediction Kleitman made fifty years ago. We also characterise all families of $a$ subsets with the smallest number of chains of length $k$ for all $a$ for which this smallest number is positive. Our argument is inspired by an elegant probabilistic lemma from a recent paper of Noel, Scott, and Sudakov, which in turn can be traced back to Lubell's proof of Sperner's theorem., Comment: 13 pages

A set A of non-negative integers is called a Sidon set if all the sums , with and a1, , are distinct. A well-known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erdős, Singer and Turan from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non-differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erdős (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that ifp(n) ≫ (logn/n)1/2, then asymptotically almost surely every subgraph ofG(n,p) with minimum degree at least (2/3 +o(1))npcontains a triangle packing that covers all but at mostO(p−2) vertices. Moreover, the assumption onpis optimal up to the (logn)1/2factor and the presence of the set ofO(p−2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

We prove that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n,p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 - α)n vertices, even after we delete a (1/2 - γ)-fraction of the edges incident to each vertex. The proof uses Szemeredi's regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.

An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty induced subgraph of size greater than c log n. Erdős, Faudree and Sós conjectured that every c-Ramsey graph with n vertices contains Ω(n5/2) induced subgraphs, any two of which differ either in the number of vertices or in the number of edges, i.e., the number of distinct pairs (|V(H)|, |E(H)|), as H ranges over all induced subgraphs of G, is Ω(n5/2). We prove an Ω(n2.3693) lower bound.

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i.e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and \Delta, we denote by T(n,\Delta) the class of all n-vertex trees with maximum degree at most \Delta. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is T(n,\Delta)-universal or, in other words, contains every spanning tree of maximum degree at most \Delta. Our methods also apply to the case when \Delta is some function of n. The result has a few very interesting implications. Most importantly, we obtain that the random graph G(n,p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded degree spanning (i.e., n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a constant. Moreover, a corresponding result holds for the random regular graph of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta, we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs with clique number at most five. Finally, we show robustness of random graphs with respect to being universal for T(n,\Delta) in the context of the Maker-Breaker tree-universality game., Comment: 25 pages

Many important theorems in combinatorics, such as Szemer\'edi's theorem on arithmetic progressions and the Erd\H{o}s-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called sparse random setting. This line of research culminated recently in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well-known conjecture of Kohayakawa, \L uczak and R\"odl, a probabilistic embedding lemma for sparse graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, and obtain their natural counting versions, which in some cases are considerably stronger. We moreover prove a sparse version of the Erd\H{o}s-Frankl-R\"odl Theorem on the number of H-free graphs and extend a result of R\"odl and Ruci\'nski on Ramsey properties in sparse random graphs to the general, non-symmetric setting. We remark that similar results have been discovered independently by Saxton and Thomason, and that, in parallel to this work, Conlon, Gowers, Samotij and Schacht have proved a sparse analogue of the counting lemma for subgraphs of the random graph G(n,p), which may be viewed as a version of the K\L R conjecture that is stronger in some ways and weaker in others., Comment: 42 pages, in this version we prove a slightly stronger variant of our main theorem

Given a graph $G$ and positive integers $n$ and $q$, let ${\bf G}(G;n,q)$ be the game played on the edges of the complete graph $K_n$ in which the two players, Maker and Breaker, alternately claim $1$ and $q$ edges, respectively. Maker's goal is to occupy all edges in some copy of $G$; Breaker tries to prevent it. In their seminal paper on positional games, Chvátal and Erdős proved that in the game ${\bf G}(K_3;n,q)$, Maker has a winning strategy if $q < \sqrt{2n+2}-5/2$, and if $q \geq 2\sqrt{n}$, then Breaker has a winning strategy. In this note, we improve the latter of these bounds by describing a randomized strategy that allows Breaker to win the game ${\bf G}(K_3;n,q)$ whenever $q \geq (2-1/24)\sqrt{n}$. Moreover, we provide additional evidence supporting the belief that this bound can be further improved to $(\sqrt{2}+o(1))\sqrt{n}$.

Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. The theorem of Schacht can be applied in a more general setting and yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdos and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting that is somewhat more general and stronger than the one obtained by Conlon and Gowers. We then use this theorem to derive several new results, among them a random version of the Erdos-Simonovits stability theorem for arbitrary graphs. The main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest., Comment: 19 pages; slightly simplified proof of the main theorem; fixed a few typos

For fixed graphs F1,…,Fr, we prove an upper bound on the threshold function for the property that G(n, p) → (F1,…,Fr). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.