Covering random graphs with monochromatic trees

Abstract

Given an r-edge-coloured complete graph Kn, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r-edge-coloured random graph G(n, p).Recently, Bucić, Korándi and Sudakov established a connection between this problem and a certain Hellytype local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-colour case by showing that (log n/n) 1/4 is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonça, Mota and Schülke. Our approach also allows us to determine the answer in the general r-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi and Sudakov.

Large monochromatic components in colorings of complete hypergraphs

“…Conjecture 5.2. Λ(2, 3, 2, 3) = 3 8 . The heuristic justification for this conjecture is as follows: As mentioned in Remark 4.1, a more careful argument can seemingly improve the lower bound on Λ(2, 3, 2, 3) to at least z ≈ 0.318.…”

“…In the other direction, he gave a construction showing that this bound is tight when r − 1 is a prime power and n is a multiple of (r − 1) 2 . Since then, many further extensions have been considered, including ones in which K n is replaced with a graph of high minimum degree [12], a nearly-complete bipartite graph [6], or a sparse random graph [1,3]; the large monochromatic component can also be taken to have small diameter [18].…”