29 problems

Let $n\geq 2k$ and let $\mathcal{F},\mathcal{G}\subseteq\binom{[n]}{k}$ be cross $1$-intersecting families. The natural parameter measuring their sizes is…

Let $G=(V,\mathcal{E})$ be a graph and let $\mathcal{I}_G$ be its hereditary family of independent sets. Write $\mathcal{I}_G^{(r)}$ for the independent sets of size $r$, and let…

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

Let $P_\ell(n)$ denote the maximum of $|\mathcal{A}||\mathcal{B}|$ over pairs of families $\mathcal{A},\mathcal{B}\subset 2^{[n]}$ such that $|A\cap B|=\ell$ for no…

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 $m\geqslant r+2$,

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 $n$ and $k$ satisfy $n>2k\geq4$. For disjoint, cross-intersecting families $\mathcal A,\mathcal B\subset\binom{[n]}{k}$, meaning that $A\cap B\neq\emptyset$ for every…

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…

For a hereditary family $\mathcal{H}$, write $\mathcal{H}^{(r)}$ for its $r$th level and define $\eta(r,s,t)$ to be the least integer $n$ such that every hereditary family with…

Let $\mathcal{H}$ be a hereditary family, and let $\mathcal{H}^{(r)}(I)=\{A\in\mathcal{H}^{(r)}:I\subseteq A\}$ denote the $r$-star centered at $I$. Let…

Sharp-threshold conjecture. The maximum-product conjecture holds with

Maximum-product conjecture. There exists an integer $n^*_{\rm P}(r,t)$ such that, whenever $\mu(\mathcal{H})\geq n^*_{\rm P}(r,t)$, there is some $T\in\mathcal{H}^{(t)}$ for which

Equality conjecture for cross-intersecting families. If $\mu(\mathcal{H})>2r$, then equality in

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…

Let $M_{m,r}$ and $M_{n,s}$ be the multiset families used in the paper, and let $\operatorname{S}_A$ denote the support of a multiset $A$. For $r,t,i,j$ in the permitted ranges, de…

Let ${\bf c}=(c_1,\dots,c_m)$ and ${\bf d}=(d_1,\dots,d_n)$ be IP sequences with $c_1\geq2$ and $d_1\geq2$. Let $r\in[m]$, $s\in[n]$, and $t\in[\min\{r,s\}]$. For $r,t,i,j$ in the…

For $n\in\mathbb{N}$ and $r,t,i,j\in\{0\}\cup[n]$ with $1\leq t\leq r$ and $t+i+j\leq n$, define … Let $m,n\in\mathbb{N}$, $r\in[m]$, $s\in[n]$, $t\in[\min\{r,s\}]$,…

Let ${\bf c}=(c_1,\dots,c_m)$ and ${\bf d}=(d_1,\dots,d_n)$ be IP sequences. Let $r\in[m]$, $s\in[n]$, and $t\in[\min\{r,s\}]$. Let $\mathcal{A}\subseteq\mathcal{S}_{{\bf c},r}$ an…

Let $1 \leq t \leq n$, $T=[t]$, and let $\mathcal{G}$ and $\mathcal{H}$ be non-empty compressed hereditary subfamilies of $2^{[n]}$. Let $g$ and $h$ be positive weight functions on…

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