431 problems

Frankl's conjecture. There exists a constant $c$ such that, whenever $k>c d^2$,

Brown–Erdős–Sós $(k+3,k)$-conjecture.

Let $[n]=\{1,2,\ldots,n\}$, and let $\mathbf{S}_k(n)$ denote the set of $k$-subsets of $[n]$. A standard representation represents each subset by an ordering of its elements. When…

Let $X$ be a finite nonempty set, let $\mathcal{F}\subseteq 2^X$ be a nontrivial monotone property, and define … and … where $w(G,p)=\sum_{S\in G}p^{|S|}$ and…

For a set of $n$ points in the unit disk, let the minimum area be the smallest area of a triangle determined by three of the points. Heilbronn's conjecture. This minimum is always…

Let $U_r(n,r)$ denote the maximum number of edges in an $n$-vertex $r$-uniform $r$-union-free hypergraph. Füredi–Ruszinkó conjecture. … This predicts that the nearly quadratic lowe…

Let $H$ be a hypergraph with a prescribed degree sequence, and consider the switch Markov chain on the realizations of that degree sequence, whose transitions apply switch operatio…

For every integer $r\ge 2$, let … Here $\pi(\mathcal{F})$ denotes the Turán density of the family $\mathcal{F}$. Chung and Graham's rationality conjecture. The set…

For a fixed $k\geq 2$, let $G^k_{n,p}$ be the random $k$-uniform hypergraph, and let $H$ be a fixed $k$-graph. The $k$-graph analogue replaces graphs by $k$-graphs, $G_{n,p}$ by…

For integers $r,k\geq 3$, let $C_k^r$ be the loose $k$-cycle in an $r$-uniform hypergraph: its edges $e_1,\dots,e_k$ satisfy $|e_i\cap e_{i+1}|=1$ cyclically, and every other pair…

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…

The $P_4^r$ upper-bound conjecture.

Füredi's conjecture. One has

Mubayi–Wang conjecture. For all $k,\ell\geq 3$,

Let $[n]=\{1,2,\ldots,n\}$, let $\binom{[n]}{k}$ denote the family of all $k$-subsets of $[n]$, and let $\mathcal{F}\subseteq\binom{[n]}{k}$ be a $k$-uniform hypergraph. Its matchi…

Feige's even-cover conjecture. Every such hypergraph contains an even cover of size $O(\ell \log n)$.

Let $\eta>0$ and let $r$ be a positive integer. For each $r$, an $r$-hyperedge coloring of the complete $r$-uniform hypergraph $\mathcal{K}_n^r$ assigns one of $r$ colors to every…

Let $\mathcal{H}$ be a hypergraph on $n$ vertices, and let the degree of a vertex in a triangulation be the number of incident faces. Bounded-degree sphere triangulation conjecture…

Let $k\geq 2$ and consider a finite complete $k$-uniform hypergraph, meaning that every $k$-element subset of its vertex set is an edge. An edge-colouring assigns a colour to each…

Győri–Katona–Lemons conjecture. The upper bound should have the same form as in the case $k>r+1$, namely

Let $\mathcal{H}$ be an $r$-uniform hypergraph on $n$ vertices, and let $\nu(\mathcal{H})$ denote its matching number. Define $\mathcal{H}_1$ as the hypergraph consisting of all…

Illingworth–Lang–Müyesser–Parczyk–Sgueglia's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ has a spanning tight component.

Georgakopoulos–Haslegrave–Montgomery–Narayanan's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ contains a spanning copy of a $(k-1)$-sphere.

A $k$-graph is a hypergraph whose edges have size $k$. It is linear if every pair of distinct edges shares at most one vertex, and a loose hypertree is a connected linear $k$-graph…

Let $r,q\in\mathbb{N}$, and partition the $r$-subsets of a $(2r+q-1)$-element set into $q$ classes. Kneser's conjecture. One of the classes contains two disjoint $r$-subsets. This…