Odd-Ramsey numbers of complete bipartite graphs

TL;DR

This paper investigates the odd-Ramsey numbers of complete bipartite graphs, providing exact solutions for spanning cases, linking the problem to coding theory, and establishing bounds for subgraph cases.

Contribution

It completely resolves the odd-Ramsey problem for all spanning complete bipartite graphs and connects it to linear binary codes, deriving asymptotically tight bounds.

Findings

Resolved the odd-Ramsey problem for all spanning complete bipartite graphs.

Established a connection between the problem and maximum dimension of certain linear binary codes.

Provided bounds for odd-Ramsey numbers of fixed, non-spanning bipartite subgraphs.

Abstract

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in \mathcal{H}$ intersects some colour class in an odd number of edges. In this paper, we focus on complete bipartite graphs. First, we completely resolve the problem when $\mathcal{H}$ is the family of all spanning complete bipartite graphs on $n$ vertices. We then focus on its subfamilies, that is, $\{K_{t,n-t}\colon t\in T\}$ for a fixed set of integers $T\subseteq [\lfloor n/2 \rfloor]$. We prove that the odd-Ramsey problem is equivalent to determining the maximum dimension of a linear binary code avoiding codewords of given weights, and leverage known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs.