On the Stability of the Independence Number of the Distance Graph G(n,r,<s) for Growing r and s

Abstract

We consider a random subgraph G_p(n,r,<s) of the complete distance graph G(n,r,<s) whose vertices are the r-element subsets of the set {1,...,n} and whose edges join pairs of subsets that intersect in fewer than s elements; each edge survives independently of the others with probability p. The stability of the Erdos-Ko-Rado theorem is the event that the independence number of the random subgraph equals that of the original graph, N = C_{n-s}^{r-s}. For fixed r and s, two-sided bounds for the critical probability, whose ratio depends on r and s, were obtained by P. A. Ogarok and A. M. Raigorodskii. In this paper we establish the order of the critical probability for growing r=r(n) and s=s(n): if min{C_r^s r^2, s r^3} = o(n), then the upper and lower bounds differ only by an absolute constant factor. The proof is elementary; only one of the two branches of the growth condition uses the complete nontrivial-intersection theorem of Ahlswede-Khachatrian.