Ahanjideh–Ekim–Yıldız general formula for triangle-free extremal graphs in terms of $Z(d)$

Let $d$ and $m$ be natural numbers, let $Z(d)$ be the parameter defined by the largest matching number of the relevant factor-critical triangle-free graph, and write

$$m=kZ(d)+r,\qquad 0\leq r<Z(d).$$

Let $f_{\vartriangle}(d,m)$ denote the maximum number of edges in a triangle-free graph with maximum degree at most $d$ and matching number at most $m$.

General formula in terms of $Z(d)$.

$$f_{\vartriangle}(d,m)= \begin{cases} dm+k\left\lfloor d/2\right\rfloor,& r<d,\\ dm+k\left\lfloor d/2\right\rfloor+r-d+1,& r\geq d. \end{cases}$$

The formula is presented as a consequence of the intermediate-value conjecture and would extend the theorem to all integers $d$ and $m$. Thus it remains conjectural whenever that preceding conjecture is unresolved.