The early evolution of the H-free process

Abstract

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page

Constructions in Ramsey theory

“…Let K 3 4 \ e denote the 3-uniform hypergraph on four vertices, obtained by removing one edge from K 3 4 . A simple argument of Erdős and Hajnal [12] implies r(K 3 4 \ e, K 3 n ) < (n!)…”

“…Here we study the off-diagonal Ramsey number, that is, r k (s, n) with k, s fixed and n tending to infinity. It is known that for fixed s ≥ k + 1, r 2 (s, n) grows polynomially in n [1, 2,3] and r 3 (s, n) grows exponentially in a power of n [6]. In 1972, Erdős and Hajnal [12] raised the question of determining the correct tower growth rate for r k (s, n).…”

Ramsey-type results for semi-algebraic relations

“…Then, by applying the induction hypothesis on (T , E) and (T i , E) for each i and by (6), there exists a subset P ⊂ T such that P 3 ∩ E = ∅ and |P | ≥ e α (r+s) (log N α 2 /18 ) e α (r+s) (log N α )…”

“…The off-diagonal Ramsey numbers, i.e., R k (s, n) with s fixed and n tending to infinity, have been intensively studied. For example, it is known [1,5,6,20] that R 2 (3, n) = Θ(n 2 / log n) and, for fixed s > 3,…”

Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs

“…For example, Erdős [8] conjectured that r(C 4 , K m ) = O(m 2− ) for some absolute constant > 0, and this conjecture is still open. The current best upper bound is an unpublished result of Szemerédi which was reproved by Caro, Rousseau, and Zhang [7] where they showed that r(C 4 , K m ) = O(m 2 / log 2 m) and the current best lower bound is Ω(m 3/2 / log m) by Bohman and Keevash [5]. In sharp contrast, for three colors Alon and Rödl [2] determined r(C 4 , C 4 , K m ) up to a poly-log factor and found the order of magnitude of r k (C 4 ; K m ) for k ≥ 3.…”