Property B: Two-Coloring Non-Uniform Hypergraphs

Abstract

The following is a classical question of Erdős (Nordisk Matematisk Tidskrift, 1963) and of Erdős and Lovász (Colloquia Mathematica Societatis János Bolyai, vol. 10, 1975). Given a hypergraph ℱ with minimum edge-size k, what is the largest function g(k) such that if the expected number of monochromatic edges in ℱ is at most g(k) when the vertices of ℱ are colored red and blue randomly and independently, then we are guaranteed that ℱ is two-colorable? Duraj, Gutowski and Kozik (ICALP 2018) have shown that g(k) ≥ Ω(log k). On the other hand, if ℱ is k-uniform, the lower bound on g(k) is much higher: g(k) ≥ Ω(√{k / log k}) (Radhakrishnan and Srinivasan, Rand. Struct. Alg., 2000). In order to bridge this gap, we define a family of locally-almost-uniform hypergraphs, for which we show, via the randomized algorithm of Cherkashin and Kozik (Rand. Struct. Alg., 2015), that g(k) can be much higher than Ω(log k), e.g., 2^Ω(√{log k}) under suitable conditions.