Disproof of a packing conjecture of Alon and Spencer. Issue 3 (29th July 2019) - Acan, Hüseyin
Abstract : A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph G n, 1/2 typically admits a covering of a constant fraction of its edges by edge‐disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for k ≪ n and A 1, …, A t chosen uniformly and independently from the k ‐subsets of {1, …, n }, what can one say about P ( | A i ∩ A j | ≤ 1 ∀ i ≠ j ) ? Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.