24 problems

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.

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

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…

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 $S$ 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 $n$ 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…

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

Nonbipartite Ramsey conjecture.

A hypertree is a $3$-uniform hypergraph in which every two vertices are connected by a unique path. A Steiner triple system is a $3$-uniform hypergraph in which every pair of verti…

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…

Let $S$ be the set of positive integers for which Steiner triple systems of order $v$ exist. For an Steiner triple system of order $v$, let the number of its subsystems of order…

A hypertree is a connected, simple $3$-uniform hypergraph in which every two vertices are joined by a unique path. A Steiner triple system is a $3$-uniform hypergraph in which ever…

High-girth triangle-decomposition conjecture. For every fixed $g$, every sufficiently large $K_3$-divisible complete graph $K_n$ has a $K_3$-decomposition with girth at least $g$.

Hesse configuration conjecture. Aside from the Hesse configuration, there are no complex Sylvester–Gallai configurations where each line passes through exactly three points.

Let $\mathcal{S}_n$ be the family of Steiner triple systems on $n$ points. For a Steiner triple system $S$, let $\alpha^*_3(S)$ denote the maximum size of a 3-independent set, and…

A Steiner triple system on $n$ vertices is a 3-uniform hypergraph in which every pair of vertices is contained in exactly one edge. A hypertree is a connected, simple 3-uniform hyp…

A Steiner Triple System is a combinatorial design consisting of triples on a vertex set such that every pair of vertices lies in exactly one triple. Its girth is the smallest integ…

Asymptotic rarity conjecture. The proportion of Steiner triple systems of order $v$ whose chromatic index is at least $m(v)+2$ tends to $0$ as $v\to\infty$.

Uniform domination conjecture. There exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that, if $D\in STS(v)$, then

Reconstruction claim. There are two nonisomorphic $\binkonf(n,1)$-configurations, each with $m$ complete subgraphs $K_n$, such that the corresponding complementary subconfiguration…

Let $STS(n)$ be the number of Steiner triple systems on $n$ points and let $F(n)$ be the number of 1-factorizations of the complete graph $K_n$. Wilson's joint asymptotic conjectur…

Nonexistence conjecture. It has no extended colorings for every $h\geq 1$.