Clique number and triangle densities in $C_4$-free graphs
For a $C_4$-free graph $G$ on $n$ vertices --- one with no induced cycle on four vertices --- we study the two-sided extremal problem for the triangle density $τ$: How large and how small can $τ$ be for given edge density $\varepsilon$ and clique-number density $κ= ω(G)/n$? We give lower and upper bounds for $τ$ in terms of $κ$ and $\varepsilon$. The two bounds sandwich $τ$, and their compatibility forces a lower bound for $κ$ in terms of $\varepsilon$. When the clique complex of $G$ is $2$-Leray over a field $\Bbbk$, the resulting bound on the clique-number density lies between the previous best $C_4$-free bound and the sharp chordal bound. It improves on the former {\it for every} $\varepsilon \in (0,1)$. The lower bound is elementary. The upper bound is homological, obtained by passing to the Stanley--Reisner ring of the clique complex. When the complex is $2$-Leray, its Betti table has at most two linear strands. The two first entries in the first strand encode edge and triangle densities, and the strong structural form of a Boij--Söderberg decomposition constrains what these entries can be, yielding the upper bound. For $2$-Leray graphs with no holes in the range $[4,g]$ we give a conjecturally sharp bound. We further ask questions concerning the triangle bound for any $C_4$-free graph.