3 problems

Let $ex(n;C_{k+1};k)$ denote the maximum number of directed copies of the cycle $C_{k+1}$ in a $k$-geodetic digraph of order $n$. The asymptotic cycle-count conjecture asserts that…

Independence conjecture. For $b_r$-regular permutations, any two cycle indicators are independent if and only if they are $r-1$-separated.

Minimum-cycle conjecture. If $\chi(G)>k$, then $G$ contains at least