Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 13, 2025, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Kalina Petrova (ISTA)

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 every graph 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. 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. This is joint work with Simona Boyadzhiyska, Shagnik Das, and Thomas Lesgourgues.

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