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Apply the Shearer independence bound for a triangle-free graph. It states that an $n$-vertex triangle-free graph of maximum degree at most $d$ has an independent set of sizefor an absolute $c>0$. The result is immediate for bounded $d$ after reducing $c$, while the theorem gives the asserted logarithmic gain for $d$ large. Thus

For completeness, the key input in Shearer's proof is to expose a random independent set one degree scale at a time. Triangle-freeness makes every neighbourhood independent, so conditioning on earlier choices creates no edges inside the available neighbours. The expected gain at degree scale $j$ is $\Omega (n_j/d)$; summing over the $\Theta (\log d)$ nonempty scales gives $\Omega (n\log d/d)$. The entropy, or hard-core-model, form of the argument makes this scale calculation rigorous without losing vertices counted at adjacent scales.

The blue hypergraph in the question is the three-edge hypergraph $K_4^{(3)-}$. The Erdős-Hajnal bound for the three-edge hypergraph on four vertices states that every $K_4^{(3)-}$-free three-uniform hypergraph on $N$ vertices has an independent set of size at leastIts proof exposes vertices successively and studies their link graphs. A blue triangle in a link is exactly a blue $K_4^{(3)-}$, so all links are triangle-free; applying the Shearer independence bound for a triangle-free graph in dyadic degree ranges either adds vertices to the independent set or leaves a reservoir whose logarithm decreases by only $O(\log \log N)$ per selected vertex. Iteration yields the displayed lower bound.

Now take $N=k^{Ck}$. For sufficiently large $C$,Thus, if there is no blue copy of $H$, the blue hypergraph has an independent $k$-set, which is a red $K_k^{(3)}$. Therefore