2,325 problems

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

Let $M_3(n)$ denote the maximum size of a family $\mathcal{F}\subseteq\binom{[n]}{4}$ with $\operatorname{VC}(\mathcal{F})\le 3$. The paper establishes the lower bound … for every…

Erdős–Simonovits' conjecture. If $e_F\geq e_{F'}$, then, for every graph $G$,

For a $3$-uniform hypergraph $F$, let $\operatorname{ex}_3(n,F)$ be the maximum number of edges in an $F$-free $3$-graph on $n$ vertices, and define its Turán density by … Let…

Let $G$ be a graph. The triangle packing number $\nu(G)$ is the maximum number of edge-disjoint triangles in $G$, and the triangle cover number $\tau(G)$ is the minimum number of e…

Brown–Erdős–Sós $(k+3,k)$-conjecture.

Ryser–Brualdi–Stein conjecture. Any proper edge-coloring of $K_{n,n}$ using $n$ colors contains a rainbow matching of size $n-1$, with a rainbow matching of size $n$ existing whene…

Erdős–Faber–Lovász conjecture. The vertices of $\mathcal{H}$ can be colored properly with $n$ colors, so that no edge contains two vertices of the same color.

Let $K_4^{(3)}$ be the complete $3$-uniform hypergraph on $4$ vertices, and let $K_4^-$ be its $3$-edge subgraph. Write $\pi(K_4^-)$ for the Turán density of $K_4^-$. Frankl–Füredi…

Erdős Matching Conjecture. Let $n\geq (k+1)r$. If $H$ is an $M_{k+1}$-free $r$-graph on $n$ vertices, then

Manickam–Miklós–Singhi conjecture. There exist at least $\binom{n-1}{k-1}$ subsets $A\subseteq [n]$ such that $|A|=k$ and

Bollobás's conjecture. Then $T\subset G$.

Let $S_n$ be the symmetric group, and let $t$-intersecting mean that any two permutations agree on at least $t$ points. For a family $F\subset S_n$, write $|F|$ for its cardinality…

Let $G$ be a graph, and let $\mu(G)$ be the minimum size of a maximal independent set in $G$. For an integer $k$, say that $G$ is $k$-EKR if no intersecting family in the family of…

Let $F$ be a finite graph. For an integer $k\geq 2$, call $F$ $k$-common if the minimum asymptotic density of monochromatic copies of $F$ in a $k$-edge-colouring equals the expecte…

Complete-graph extremality conjecture. For every graph $G$ with $n$ vertices,

A topological graph is a graph drawn in the plane with vertices represented by points and edges by simple Jordan arcs; it is $k$-quasi-planar if it contains no $k$ pairwise crossin…

A thrackle is a simple topological graph in which every pair of edges that do not share a vertex crosses. Conway's thrackle conjecture. Every thrackle with $n$ vertices has at most…

Let $D$ be a digraph. For an integer $s\ge 1$, call a set $S\subseteq V(D)$ an $s$-source set if … where $N^-(S)$ is the set of external in-neighbors of $S$. For an integer…

Ge, Xu, and Zhang's conjecture. For every fixed $t$,

A tree is a connected graph with no cycles, and a labelling of a graph with $m$ edges is an injective map from its vertices to the positive integers. Such a labelling is graceful i…

Lemmens–Seidel's conjecture.

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{A}$ be a family of subsets of $[n]$ or a family of labelled graphs on $n$ vertices. In the first setting, require that every pair of set…

Pósa-type conjecture for triple systems. For every $\varepsilon>0$, there is $n_0$ such that every $3$-uniform hypergraph $G$ of order $n\geq n_0$ satisfying

For positive integers $I$ and $J$, let $H(I,J)$ be the least $n$ for which there are $I$ disjoint $J$-mark Golomb rulers in $\{1,\ldots,n\}$. Regularity conjecture for $H(I,I+2)$.…