1,300 problems

Let $G_n$ be the partition graph, let $\mathrm{Ax}_n$ and $\mathrm{Sp}_n$ denote its axial and spinal vertex sets, and let…

Kahn's conjecture. If $|X|=|Y|=|Z|$, then

Let $X$, $Y$, and $Z$ be upward closed subsets of $Q_n$. It is conjectured that … Notation: the poset $Q_n$ is the power set of a set with $n$ elements, ordered by subset relation.…

For every constant $c>0$, there exists a function $f_c(n)=o(n)$ such that every graph $G$ on $n$ vertices with independence number $\alpha(G)\ge cn$ satisfies $h(G)\le f_c(n)$, whe…

For every $3$-uniform hypergraph $F$, $\pi_{\mathrm{co}}(F)=0$ if and only if $F$ is layered and $\pi(F)=0$, where $\pi(F)$ is the uniform Turán density and…

For integers $k\ge 2$, $s\ge 1$, and $n\ge (s+1)k$, every family $\mathcal{F}\subseteq\binom{[n]}{k}$ with matching number $\nu(\mathcal{F})\le s$ satisfies …

For each positive integer $n$, let $U(n)$ be the maximum, over all sets $P\subseteq\mathbb{R}^2$ with $|P|=n$, of the number of unordered pairs $\{p,q\}\subseteq P$ satisfying…

For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, define $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$, and let $f(n,4)$ be the minimum possible value of $|E(G)|$ o…

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The sources identify Erdős problem $\#608$ as an extremal problem involving a claimed threshold of $2/9$, but do not provide the underlying objects or define the extremal quantity…

For every integer $n\ge 1$ and every integer $s$ satisfying $1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}$, let $\mathcal F$ be a family of even-sized subsets of $[n]$ w…

For every $\varepsilon>0$, there exists a constant $C_0>0$ such that, for every digraph $H$ with $h$ arcs and no isolated vertices, every $n$-vertex digraph $D$ satisfying…

For every $d>0$ and every $\alpha>0$, do there exist $\rho>0$ and $n_0$ such that every $(\rho,d)$-dense $3$-uniform hypergraph $H$ on $n\ge n_0$ vertices with minimum codegree…

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$. The family $\mathcal{F}$ is $t$-intersecting if $|A\cap B|\geq t$ for all $A,B\in\mathcal{F}$, and it is $k$-Spern…

Jump threshold conjecture. All numbers in $[0,4/9)$ are jumps and all numbers in $[4/9,1)$ are non-jumps.

Let $G$ be an extremal $\ell$-uniform hypergraph for Problem 13 on $n$ vertices, and let $K_{t+1}^{\ell}$ denote the complete $\ell$-uniform hypergraph on $t+1$ vertices. An isolat…

Extremal characterization conjecture. $f_r(s)$ achieves the maximum for either $\mathcal{K}_{s+r-1}^r$, or the hypergraph $\mathcal{G}$ on $s+r$ vertices obtained by this minimum-e…

For integers $r\ge 3$ and $r-2\ge t\ge 1$, let $\iota_0(t,n;r)$ denote the maximum size of a $t$-intersecting family in an $r$-partite $r$-uniform hypergraph with all parts of size…

Cycle-with-chords conjecture. If $G$ has average degree at least $C\log\log(n)$, then $G$ contains a cycle $C$ on $[202~\ell$ vertices with at least $[203~\ell/2$ chords, for some…

Let $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ denote its power set. For a family $\mathcal{F}\subset 2^{[n]}$, write … and define its diametral overflow by … where…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no collection of $k$ edges spanning at most $s$ vertices. For positive intege…

Conjecture for totally chain-intersecting families. If $\mathcal F$ is a totally $(p,q)$-chain intersecting family and $p\geq q$, then

The 3/8 conjecture.

Affine-vector-space minimal-size conjecture. The minimal size of a tight irreducible affine vector space partition of length $n$ is

Maximum-minimum-dimension conjecture. Every tight irreducible subcube partition $F$ of length $n$ satisfies