7 problems

Let $G$ be a graph with chromatic number $k$, let $G^r$ denote its $r$-uniform power, and let $m'(G)$ be the number of edges between a specified pair of color classes of a $(k-1)$-…

Let $F$ be a graph, let $\rchi(F)$ denote its chromatic number, let $q(F)$ be the minimum number of red vertices in a strong red-blue coloring of $F$, and let $H_i$ be the 3-unifor…

Bridgelessness characterization.

Hilton–Milner codegree-squared-sum conjecture. For $k\ge3$ and $n>2k$,

Exact codegree-squared-sum conjecture. If $t\ge1$ and $n\ge(t+1)(k-t+1)$, then

The conjecture on larger path lengths. The exact result should hold for much larger $\ell$, possibly as far as

The conjecture on linear paths in triple systems. A similar result to the stated exact formulas for $k\geq 4$ should hold for $k=3$: for positive integers $t$, the corresponding ex…