A Completion of the Proof of the Edge-statistics Conjecture

Abstract

For given integers k and ℓ with 0 < ℓ < k 2 , Alon, Hefetz, Krivelevich and Tyomkyn formulated the following conjecture: When sampling a k-vertex subset uniformly at random from a very large graph G, then the probability to have exactly ℓ edges within the sampled k-vertex subset is at most e −1 + o k (1). This conjecture was proved in the case Ω(k) ≤ ℓ ≤ k 2 − Ω(k) by Kwan, Sudakov and Tran. In this paper, we complete the proof of the conjecture by resolving the remaining cases. We furthermore give nearly tight upper bounds for the probability described above in the case ω(1) ≤ ℓ ≤ o(k). We also extend some of our results to hypergraphs with bounded edge size.

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

“…On the other hand, the two results in the last paragraph point in the opposite direction: there are many different possibilities for the numbers of edges in induced subgraphs. In connection with some recent work on anti-concentration of edge-statistics (see [2,22,27,30]), Kwan, Sudakov and Tran [27] asked about anti-concentration of the edge distribution in Ramsey graphs. Specifically, for an n-vertex C-Ramsey graph, let X be the number of edges induced by a uniformly random set of (say) n/2 vertices.…”

The edge-statistics conjecture for $\ell \ll k^{6/5}$

“…Even more recently, and independently of our own work, Fox and Sauermann [3] also gave a proof of Conjecture 1.1. The proof given here has the advantage that it is considerably shorter than the one in [3].…”

“…Even more recently, and independently of our own work, Fox and Sauermann [3] also gave a proof of Conjecture 1.1. The proof given here has the advantage that it is considerably shorter than the one in [3]. However, [3] contains some stronger bounds in certain ranges of ℓ (e.g., it is shown that in fact ind(k, ℓ) = o k (1) when ω k (1) ℓ o k (k)), as well as results for the analogous problem in hypergraphs and other related results.…”

Proof of a conjecture on induced subgraphs of Ramsey graphs

“…For example, is this random variable anticoncentrated? For general graphs this question was first studied by Alon, Hefetz, Krivelevich and Tyomkyn [3] (see [28,23,29] for further work). Regarding Ramsey graphs, as we recently proposed in a paper with Tuan Tran [28], could it be true that in any O(1)-Ramsey graph G, if A is a uniformly random set of n/2 vertices, then Pr(e(G[A]) = x) = O(1/n) for all x?…”