Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nešetřil and Rödl proved that for every graph $G$, there exists a graph $F$ such that $F\to G$ and $\omega (F)=\omega (G)$ (where $\omega$ is the size of the largest clique in the graph).

My question is an extension of that. Is there a graph $F$ so that $F \to P_{2n}$ but so $F$ is $C_{2n+1}-$free? This is trivial for $n=1$. For $n=2$, the graph $F:= C_6\cup \{(x,v_i)|i=1,3,5\}$ works (where $v_1,\dots,v_6$ are the vertices of the $C_6$ and $x\not \in C_6$).

The motivation is that if this is true, I can construct another graph with a special property (using this lemma) that will help me solve a much more interesting problem (depending on how interesting this itself is!)

So my question is, does this generalize to all $n$? It seems that it should.