Ahanjideh–Ekim–Yıldız formula for maximum edges in triangle-free graphs

For a positive integer $d$, let $Z(d)$ denote the relevant extremal matching-number threshold, and let $f_{\Delta}(d,m)$ be the maximum number of edges in a triangle-free graph with matching number $m$ and maximum degree bounded in the setting of the source. For natural numbers $d\geq 2$ and $m$, write

$$m=kZ(d)+r.$$

Ahanjideh–Ekim–Yıldız conjecture. For all natural numbers $d\geq 2$ and $m$, we have

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

This formula extends the cases established in the source to the remaining open cases and would give the general edge-extremal function for the considered class of triangle-free graphs.