16 problems

General pinnable covering conjecture. If $A\cup B$ is pinnable, then

Pinnable covering conjecture. If $A\cup B$ is pinnable, then

Projective-plane extremal conjecture. For $m\geqslant r+2$,

Gyárfás–Lehel conjecture. Then

Let $(A_1,B_1),\dots,(A_m,B_m)$ be pairs of finite non-empty subsets of $\mathbb{N}$. Write $a_i=|A_i|$ and $b_i=|B_i|$ for $1\leq i\leq m$. Assume that … … and…

Let $\mathcal{H}$ be a hereditary family, let $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ be its levels, and suppose $\mathcal{A}\subseteq\mathcal{H}^{(r)}$ and…

Let $\mathcal{H}$ be a hereditary family, with $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ its $r$th and $s$th levels. For cross-intersecting subfamilies, let…

Cross-intersection conjecture for hereditary families. If $\mu(\mathcal{H})\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{H}^{(r)}$ are cross-intersecting, then

Kamat's conjecture. If $\mu(\mathcal{I}_G)\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{I}_G^{(r)}$ are cross-intersecting, then

Let $X_1$ and $X_2$ be disjoint sets with sizes $n_1$ and $n_2$, and let ${X_1, X_2\choose k,\ell}$ denote the family of sets $F$ satisfying $|F\cap X_1|=k$ and $|F\cap X_2|=\ell$.…

Let $1\leq t\leq r$, let $\emptyset\neq S\subseteq [r]$, and let $\mathcal{H}$ be a hereditary family with covering number $\mu(\mathcal{H})\geq (t+1)(r-t+1)$. Write…

Let $t\geq 1$, let $\mathcal{F}$ be a family, and let $m$ be a positive integer. The notation $\mathcal{S}_{\mathcal{F},m}$ denotes the associated lifted set family, and the strong…

Stability conjecture. There exists a constant $c=c(p_1,p_2)$ such that, for every pair of cross-intersecting families $U_1\subset\Omega_1$ and $U_2\subset\Omega_2$ satisfying

Borg's product conjecture. The product

Borg's stronger sum conjecture. (i) If

Borg's sum conjecture. If $k\geq n+1$, then