Odd-Ramsey numbers of complete bipartite graphs

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $H$ 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 $H\in H$ 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 $H$ is the family of all spanning complete bipartite graphs on $n$ vertices. We then focus on its subfamilies, that is, $\{K_{t,n-t}:t\in T\}$ for a fixed set of integers $T\subseteq \left[\lfloor n/2\rfloor \right]$. 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.