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Let the blue edges form a graph $B$ on $N$ vertices, and suppose there is no blue copy of $H$. For every vertex $x$, the graph $B[N_B(x)]$ has maximum degree at most one. Indeed, if some $y\in N_B(x)$ had two neighbours $z,w$ inside $N_B(x)$, thenwould be the five blue edges of a copy of $H$.

Let $d=\Delta (B)$. If $d\ge 2k$, then a maximum-degree neighbourhood, being a matching plus isolated vertices, has an independent set of size at least $d/2\ge k$. This is a red $K_k$. We may therefore assume $d<2k$. Part (b) givesIf $d<\sqrt{k}$, the greedy independent-set bound gives $\alpha (B)\ge N/(d+1)>k$ once $N=C{k}^2/\log k$. If $\sqrt{k}\le d<2k$, then part (b) giveswhen $C$ is sufficiently large. In either case there is a red $K_k$, so

For any two states $X,Y$, the closed neighbourhood of $X\cup Y$ has at most $2s(\Delta +1)\le 2n/3$ vertices. The remaining induced graph therefore has at least $n/3$ vertices and maximum degree at most $\Delta$, so the greedy independent-set bound supplies an independent set $Z$ of size $s$ there. No vertex of $Z$ is adjacent to a vertex of $X\cup Y$. Replace the elements of $X$ one at a time by the elements of $Z$, and then replace the elements of $Z$ one at a time by those of $Y$. Every intermediate set is independent, and every prescribed swap has positive transition probability. Hence the chain is irreducible.