218 problems

Erdős matching conjecture. The extremal function satisfies

Let $[n]$ be the ground set, and let a $k$-chain be a sequence of $k$ distinct subsets of $[n]$ ordered by strict inclusion. For each family size $a$, consider the centralized fami…

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…

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 $\mathcal{F}\subseteq\binom{[n]}{k}$, with $k>d$. A $d$-simplex is a collection of $d+1$ sets with empty common intersection while every proper subcollection has nonempty inter…

Frankl–Wang's $3/7$ conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting, $n>100k$, and $|\mathcal{F}|>\binom{n-3}{k-3}$, then

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$, and let $\binom{[n]}{k}$ be the family of $k$-subsets of $[n]$. Let $\mathcal F=\{F_1,\ldots,F_t\}$ be a family of subsets o…

For a finite poset $(P,\leqslant)$, let $e(P)$ be the largest integer such that the union of any $e(P)$ consecutive layers of $2^{[n]}$ is weak $P$-free, and let…

Let $n$ and $d$ be positive integers, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be a $(d+1)$-uniform family with VC-dimension at most $d$. Erdős–Frankl–Pach conjecture. When…

Füredi's conjecture. For every positive integer $n$, the Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that every chain has size eit…

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

Friedgut–Kahn–Kalai–Keller's off-diagonal conjecture. One should have

Exact EKR conjecture. Then

Ahlswede–Khachatrian conjecture. For all $n$, $k$, and $t$, a largest $t$-intersecting family is isomorphic to one of the families $\mathcal D_i$.

Let $[n]=\{1,\dots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$, and let a $d$-simplex be a collection of $d+1$ members of $\binom{[n]}{k}$ whose total inte…

Let $P$ be a finite poset. For a family $\mathcal F\subseteq 2^{[n]}$, let $La(n,P)$ be the largest size of a $P$-free family, and let $La^*(n,P)$ be the largest size of an induced…

Let $K(n,k)$ be the Kneser graph whose vertices are the $k$-subsets of $[n]$, with two vertices adjacent when the corresponding sets are disjoint. For an integer $s\geq 3$, a famil…

Let $0\le s\le d$, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be an $s$-witness family, meaning that for every $F\in\mathcal{F}$ there exists $B_F\in\binom{F}{s}$ such that…

For positive integers $I$ and $J$, an $(I,J,n)$-DGR is a collection of $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Let $H(I,J)$ be the least positive integ…

Let $\mathcal{F}$ be a $k$-graph on vertex set $[n]$. Let $\nu(\mathcal{F})$ denote its matching number and let $\tau(\mathcal{F})$ denote the minimum size of a vertex cover, meani…

Let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$. Rainbow Erdős Matching Conjecture. If $n\geq(s+1)k$ and…

Let $\mathcal{F}$ be a finite union-closed family of finite sets. Gilmer's proposed bound. Similar information-theoretic methods should lead to the existence of an element that bel…

Lexicographic initial-segment conjecture. For fixed $n>k\geq3$, if $[S_m]$ is an FC-family for some positive integer $m$ and has universe size $n$, then

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