3 problems

Let $F$ be a graph with $e(F)>1$, and let $m=|F|$. For a graph property $\mathcal P(F;1/m,\dots,1/m)$ and its weaker version \widetilde\mathcal P(F;1/m,\dots,1/m), the source con…

For a graph $G$ on $n$ vertices, write $e(G)$ for its number of edges and set … For nonempty vertex subsets $X,Y\subseteq V(G)$, let … Let $\sigma_{2}(G)$ denote the second-largest…

Three-variable kernel conjecture. If $\alpha\notin\{1/3,2/3\}$, then $f=0$ almost everywhere. If $\alpha=1/3$, then, almost everywhere,