On edges not in monochromatic copies of a fixed bipartite graph

Let $H$ be a fixed graph. Denote $f(n,H)$ to be the maximum number of edges not contained in any monochromatic copy of $H$ in a 2-edge-coloring of the complete graph $K_n$, and $ex(n,H)$ to be the {\it Tur\'an number} of $H$. An easy lower bound shows $f(n,H)\ge ex(n,H)$ for any $H$ and $n$. In \cite{KS2}, Keevash and Sudakov proved that if $H$ is an edge-color-critical graph or $C_4$, then $f(n,H)= ex(n,H)$ holds for large $n$, and they asked if this equality holds for any graph $H$ when $n$ is sufficiently large. In this paper, we provide an affirmative answer to this problem for an abundant infinite family of bipartite graphs $H$, including all even cycles and complete bipartite graphs $K_{s,t}$ for $t>s^2-3s+3$ or $(s,t)\in\{(3,3),(4,7)\}$. In addition, our proof shows that for all such $H$, the 2-edge-coloring $c$ of $K_n$ achieves the maximum number $f(n,H)$ if and only if one of the color classes in $c$ induces an extremal graph for $ex(n,H)$. We also obtain a multi-coloring generalization for bipartite graphs. Some related problems are discussed in the final section.