6 problems

Alon's non-star-forest conjecture. There is a universal constant $c>0$ such that, whenever $H$ is not a star forest, the largest $H$-intersecting family on $n$ vertices has size at…

$P_3$-umvirate conjecture. The largest $P_3$-intersecting family is a $P_3$-umvirate. This conjecture is false: Christofides constructed a $P_3$-intersecting family on $6$ vertices…

Biased extension conjecture. These biased results should extend to $1/2<p\leq 3/4$. This conjecture was disproved by Keller and Lifschitz, so the proposed extension is false.

Cross-$t$-chromatic-agreeing conjecture. The cross-$t$-chromatic-agreeing analogue of the source's theorem should hold for all $t$, with upper bound

Cross-$K_t$-intersecting conjecture. The same theorem, including its upper bound, uniqueness, and stability conclusions, should hold for all $t\geq 5$. The source proves the cases…

A cycle chain is the graph family constructed in the paper by joining cycles in a chain. A graph is prime if it has a vertex labeling by consecutive integers starting at $1$ such t…