Erdős–Simonovits conjecture for the asymptotic number of triangle- and four-cycle-free edges

Determine whether the limit of the normalized extremal function for graphs containing neither a triangle nor a four-cycle exists with value \(1/(2\sqrt{2})\), namely, whether \(\lim_{n\to\infty}\mathrm{ex}_2(n,\{C_3,C_4\})/n^{3/2}=1/(2\sqrt{2})\).

Background

The paper places this question in the general context of extremal graph theory, where one first seeks the polynomial order of an extremal function and then asks whether a precise leading asymptotic constant exists. It states that $\mathrm{ex}_2(n,\{C_3,C_4\})=\Theta(n^{3/2})$ is known, but that the sharper asymptotic constant remains conjectural.

The problem concerns the maximum number of edges in an $n$-vertex graph containing no triangle $C_3$ and no four-cycle $C_4$. The conjectured limit would identify the exact leading constant in its $n^{3/2}$-scale growth.