11 problems

Let $n$, $k$, and $t$ be positive integers with $k\geq t$ and $n\geq(t+1)(k-t+1)$. Two families $\mathcal{A},\mathcal{B}\subseteq\binom{[n]}{k}$ are cross-$t$-intersecting if…

Suda–Tanaka–Tokushige's stability conjecture. Assume

Frankl–Tokushige product conjecture. If $k_i\leq (r-1)n/r$ for every $i\in[r]$, then

Let $\widetilde{m}(n,k,k,2,2)$ be the maximum product of the sizes of two cross-intersecting $k$-uniform families, each having covering number $2$. The extension conjecture. The fo…

Let $n\geq 2k>2\ell\geq 4$. Fix sets $F_0\in\binom{[2,n]}{k}$ and $G_0\in\binom{[2,n]}{\ell}$ with $F_0\cap G_0\neq\emptyset$, and define … … Two families are cross-intersecting if…

Three-fold cross-intersection product-measure conjecture. If $\mathcal A_1,\mathcal A_2,\mathcal A_3$ are 3-cross intersecting, then

Let $\Omega$ be the parameter domain used in Theorem, and let $\mu_\alpha$ and $\mu_\beta$ denote the corresponding biased measures on $2^{[n]}$. Let $\mathcal A_j$ and…

Let $n,k,l$ be positive integers with $k+l<n<\max\{2k,2l\}$. For each $j\geq 0$, let … and … Let $M(n,k,l)$ be the maximum of $|\mathcal A||\mathcal B|$ over cross-intersecting fam…

The cross-shade asymptotic conjecture. Then

Let $C(n,k,l,t)$ be the collection of cross-$t$-intersecting pairs $(\mathcal A,\mathcal B)$, and define … For $0\le i+j\le n-t$, define…

The cross-$t$ Erdős–Ko–Rado conjecture. For fixed $k,l,t$, one has