Many important theorems and conjectures in combinatorics, such as the the- orem of Szemeredi on arithmetic progressions and the Erd˝os-Stone Theorem in extremal graph theory, can be phrased as… Expand

Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of… Expand

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… Expand

This work proves a variant of the KŁR conjecture which is sufficient for most known applications to random graphs and implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems.Expand

A slightly weaker version of Haxell's result is proved that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees.Expand

It is proved that there exists a positive constant $\varepsilon$ such that if if n / n p = n + 1, then asymptotically almost surely the random graph G(n,p) contains a collection of $\lfloor \delta(G)/2 \rfloor$ edge-disjoint Hamilton cycles.Expand

Estimating the number of Sidon sets of a given cardinality contained in [n] is based on estimating the random variable, where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m.Expand

It is shown that if p(n) ≫ (log n/n)1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p−2) vertices.Expand

This work shows that every n-vertex graph satisfying certain natural expansion properties is (n,Δ)-universal, and shows that random graphs are robustly ( n,� Δ)-universal in the context of the Maker–Breaker tree-universality game.Expand

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise… Expand