51 problems

Elliott–Rödl conjecture. Every hypertree of $(1-o(1))n$ vertices can be found in any Steiner triple system.

Erdős' conjecture. For each integer $g\geq 3$ and all sufficiently large $n\equiv 1,3\pmod 6$, there exists a Steiner triple system on $n$ elements with girth at least $g$.

A Steiner triple system of order $n$ is a $3$-uniform hypergraph on $n$ vertices in which every pair of vertices lies in exactly one edge. A matching is a set of pairwise vertex-di…

Let $(X,\mathcal B)$ be a cyclic Steiner triple system of order $v$, with $v\equiv 1\pmod 6$, and identify its point set with $\mathbb Z_v$. Partition the blocks into orbits under…

Lindner's conjecture. Any partial Steiner triple system of order $m$ can be embedded in a Steiner triple system of order $n$ whenever

Erdős meets Nash-Williams' conjecture. For every integer $g$, every sufficiently large $K_3$-divisible graph $G$ on $n$ vertices satisfying

Nine-vertex 2-core conjecture. Every sufficiently large Steiner triple system contains a 2-core on nine vertices.

A Steiner triple system (STS) is a collection of 3-element blocks on a finite point set such that every pair of distinct points lies in exactly one block. Erdős's order-21 STS conj…

Let $\theta_{\operatorname{STS}}$ denote the minimum codegree threshold for Steiner triple systems, and let $\theta_{\operatorname{STS}}^*$ denote the corresponding fractional thre…

Let $n_0\in\mathbb{N}$ and $C\in\mathbb{R}$ be constants. For every $n\geq n_0$ satisfying $n\equiv 1,3\pmod{6}$, let $\mathcal{H}=\{H_1,\dots,H_{n(n-1)/6}\}$ be a family of $3$-un…

Let a Steiner triple system of order $n$ be a $3$-uniform hypergraph on $n$ vertices in which every pair lies in exactly one edge, and let a perfect matching be a set of $n/3$ pair…

Let $S$ be a Steiner quasigroup of order $n$ selected uniformly at random. Let $\ell(S)$ be the length of its longest row cycle, and let $m(S)$ be the size of its largest proper St…

Let $k$ be a positive integer, and let $n$ be sufficiently large and satisfy the necessary divisibility conditions for the existence of a Steiner triple system of order $n$. A Stei…

Let $n$ be sufficiently large with $n\equiv 3\pmod{6}$, and let $V$ be a common $n$-vertex set. Let $\mathcal{H}=\{H_1,\ldots,H_{n(n-1)/6}\}$ be a collection of $3$-uniform hypergr…

Let $n$ be sufficiently large with $n\equiv 3\pmod{6}$, and let $H$ be a hypergraph on $n$ vertices. A resolvable Steiner triple system is a Steiner triple system whose triples can…

Let $n$ be sufficiently large with $n\text{ congruent to }1$ or $3\text{ modulo }6$, and let $H$ be an $n$-vertex $3$-uniform hypergraph. The minimum codegree threshold conjecture.…

Steiner triple code embedding-dimension conjecture. For every $d\ge 1$ there exists a Steiner triple code $\mathcal{C}$ with

Classification conjecture. These codes comprise only the classic examples of Cameron–Liebler line classes: a point, a hyperplane, and a nonincident point–hyperplane pair.

Random-hypergraph conjecture. Using a random hypergraph model, this proportion should be

Nonbipartite Ramsey conjecture.

Let $K_n$ be the complete graph on $n$ vertices, and let $\randt(K_n,p)$ be its binomial random 3-uniform triangle system. The sharp threshold is the transition scale at which the…

Let $n\equiv 1,3 \pmod 6$, and let $T_1,T_2,\ldots$ be a uniformly random ordering of the triples in $\binom{[n]}{3}$. Steiner triple system hitting-time conjecture. With high prob…

Erdős's conjecture. For any $g\in\mathbb{N}$, if $N$ is admissible and sufficiently large in terms of $g$, then there is a Steiner triple system which has no $(j,j-2)$-configuratio…

Full-configuration sufficiency conjecture. No subset of the full $m$-line configurations suffices for this purpose.

A Steiner triple system of order $n$ is a $3$-uniform hypergraph on $n$ vertices in which every pair of vertices lies in exactly one edge. Its girth is governed in particular by th…