Optimal clique-number density function for induced-$C_4$-free graphs

Determine the optimal function $f(\\varepsilon)$ such that every induced-$C_4$-free graph of edge density $\\varepsilon$ has clique-number density at least $f(\\varepsilon)$.

Background

The paper studies lower bounds on the clique-number density κ=ω(G)/n\kappa=\omega(G)/n of induced-C4C_4-free graphs in terms of the edge density ε\varepsilon. Existing results establish positive lower bounds, including κ≥(1−1−ε)2\kappa\geq (1-\sqrt{1-\varepsilon})^2, but the authors state that the optimal dependence of κ\kappa on ε\varepsilon is not known.

References

The optimal function $f$ is nevertheless unknown.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Section 1, subsection “The clique-number density problem”

We are not aware of any $C_4$-free graph violating `eq:upper, even when its clique complex has nonvanishing homology in dimension two or higher. This leads to the principal structural question raised by the paper: Does the upper triangle-density bound \eq:upper` hold for every $C_4$-free graph, without assuming that its clique complex is $2$-Leray?

eq:upper:

τ≤(3−κ2) κ ε1+2κ−κ2.\tau \leq \frac{(3-\kappa^2)\,\kappa\,\varepsilon} {1+2\kappa-\kappa^2}.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Question 1.6, subsection “The Leray hypothesis and open questions”; reiterated in Section “Further discussion”

A further open problem is whether the sandwich bound is tight.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Section “Further discussion,” paragraph following Theorem 6.1