Covering random graphs with monochromatic trees

Abstract

Given an r$$ r $$ ‐edge‐colored complete graph Kn$$ {K}_n $$ , 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$$ r $$ ‐edge‐colored random graph 𝒢(n,p) . Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly‐type 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‐color case by showing that (logn/n)1/4$$ {\left(\log n/n\right)}^{1/4} $$ is a threshold at which point three monochromatic components are needed to cover all vertices of a 3‐edge‐colored 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$$ r $$ ‐edge colored 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.