  Recently, Alon introduced the notion of an $H$-code for a graph $H$: a
collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two
members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote
the maximum possible cardinality of an $H$-code, and let
$d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on
$d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors
needed to edge-color $K_n$ so that every copy of $H$ has an odd color class.
  Motivated by this observation, we define $g(G,H)$ to be the minimum number of
colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd
color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4)=
n/2+o(n)$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was
obtained independently in arXiv:2306.14682.

Combinatorics