Given a graph $G(V,E)$ whose edges are colored in two colors: red and blue. Suppose the following two conditions hold:
• for any $S\subseteq V$, there are at most $O(|S|)$ red edges in $G[S]$
• for any $S\subseteq V$, if $G[S]$ contains no red edges, then it contains $O(|S|)$ blue edges
One can maybe consider first an easier version, if we assume that the red degree of every vertex is $O(1)$. In this case I also don't know the answer.