Extremality of clique-number density $1/4$ at edge density $1/2$

Determine whether every $C_4$-free graph with edge density asymptotically equal to $1/2$ has clique-number density at least $1/4$, equivalently whether the circulant construction gives the optimal asymptotic value at $\\varepsilon=1/2$.

Background

The circulant graphs C4k+1kC_{4k+1}^k are C4C_4-free, have edge density tending to $1/2$, and have clique-number density tending to $1/4$. The construction is extremal among regular graphs, but the paper leaves unresolved whether an arbitrary, not necessarily regular, C4C_4-free graph at this edge density can have smaller clique-number density.

References

Whether $\kappa = \tfrac14$ is optimal at $\varepsilon = \tfrac12$ over all $C_4$-free graphs remains open, and we return to this point in Section~\ref{sec:discussion}.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Example 1.5 and Section “Further discussion,” paragraph “The point $\\varepsilon=\\tfrac12$”

In particular, taking $\delta(G) \geq \tfrac n2$ yields $\omega(G) \geq \tfrac n6$, and they remark that whether the same conclusion follows from the edge-density hypothesis $\varepsilon \geq \tfrac 12$ remains open.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Section “Further discussion,” paragraph “The point $\\varepsilon=\\tfrac12$”

We do not know whether `eq:upper` is preserved under the clique-sum of $G$ with an arbitrary $C_4$-free graph, nor under the join $G\ast K_s$ that adjoins $s$ universal vertices.

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 Section “Further discussion,” final paragraph before subsection “How far the strand method reaches”

A sharper argument might retain such an entry and work in an enlarged $(m,b,t)$-space, where constraining $t$ could exclude points that the planar projection cannot, or might exploit that not every diagram admissible under Lemma~\ref{lem:homology vertices} arises from an actual clique complex. We do not know whether either route helps.

— Clique number and triangle densities in $C_4$-free graphs  (2608.19686 - Fløystad et al., 20 Aug 2026) in Section “How far the strand method reaches,” final paragraph