Resolving the Kohayakawa--Kreuter Conjecture for Families

TL;DR

This paper proves a conjecture about partitioning sparse graphs into specific subgraphs and confirms a threshold for a random graph property related to Ramsey theory for graph families.

Contribution

It fully proves the conjecture on partitioning $(a,b)$-sparse graphs and links it to the Kohayakawa--Kreuter Conjecture for Families in random graph theory.

Findings

Confirmed the conjecture on partitioning $(a,b)$-sparse graphs.

Established the threshold function for the Ramsey property in random graphs.

Connected the graph partitioning conjecture to the Kohayakawa--Kreuter Conjecture for Families.

Abstract

A graph $G$ is $(a,b)$-sparse if every nonempty subgraph $H$ satisfies $e(H) \leq a v(H) - b$. We are interested in the conditions under which an $(a,b)$-sparse graph can be partitioned $E(G) = E(G_1) \cup E(G_2)$ such that for $i \in \{1,2\}$ we have that $G_i$ is $(a_i, b_i)$-sparse. Kuperwasser, Samotij, and Wigderson conjectured that a $(m,0)$-sparse graph can be partitioned into a $(1,1)$-sparse graph and a $(m,2m-1)$-sparse graph. We prove the conjecture in full. The Kohayakawa--Kreuter Conjecture for Families claims that $n^{-1/m_2}$ is the threshold function for the random graph being Ramsey a.a.s. for graph families $\mathcal{H}_1, \ldots \mathcal{H}_r$. Kuperwasser, Samotij, and Wigderson motivated their conjecture by proving that it is sufficient to establish the Kohayakawa--Kreuter Conjecture for Families.