A variant of the Erd\H{o}s-Gy\'arf\'as problem for $K_8$

Abstract

Recently, Alon initiated the study of graph codes and their linear variants in analogy to the study of error correcting codes in theoretical computer science. Alon related the maximum density of a linear graph code which avoids images of a small graph $H$ to the following variant of the Erd\H{o}s-Gy\'arf\'as problem on edge-colourings of $K_n$. A copy of $H$ in an edge-colouring of $K_n$ is even-chromatic if each colour occupies an even number of edges in the copy. We seek an edge-colouring of $K_n$ using $n^{o(1)}$ colours such that there are no even-chromatic copies of $H$. Such an edge-colouring is conjectured to exist for all cliques $K_t$ with an even number of edges. To date, edge-colourings satisfying this property have been constructed for $K_4$ and $K_5$. We construct an edge-colouring using $n^{o(1)}$ colours which avoids even-chromatic copies of $K_8$. This was the smallest open case of the above conjecture, as $K_6, K_7$ each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of $H$, there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using $n^{o(1)}$ colours and satisfying this stronger requirement exists for all cliques $K_t$ regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for $K_4$ and $K_5$. These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of $K_4$ and $K_5$.