59 problems

Erdős–Hanani conjecture. Approximate $(n,q,r)$-Steiner systems exist.

Let $q>r\geq 2$ and let a girth-$g$ decomposition mean a $K_q^r$-decomposition containing no configuration of the forbidden size below $g$. High Girth Existence Conjecture. For eve…

Let $STS(n)$ denote the number of non-isomorphic Steiner triple systems on $n$ vertices, where $n$ is admissible, meaning $n\equiv 1,3\pmod{6}$. Wilson's conjecture. … This gives t…

Wanless's conjecture. Every partite $(d,d+1,n)$-Steiner system has a perfect matching if $n$ or $d$ is even.

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices in which each $q$-set is present independently with probability $p$. A Steiner system is an…

Let $M$ be an $n\times m$ $t$-disjunct matrix with constant column weight $b$. Set … and assume that $p$ is an integer. A Steiner system $\mathfrak{S}(n,b,p)$ is a family of $b$-su…

Let $n>q>r\geq 1$ be integers. An $(n,q,r)$-Steiner system is a collection of $q$-subsets of an $n$-element set such that every $r$-subset is contained in exactly one member. Equiv…

Let $k$ be the cardinality of a finite field, let $G=SL(2,\mathbb F_k)$, and let a $G$-space split into one orbit of size $|G|$ and $k+1$ orbits of size $1$. The SL(2) infinite-ser…

Keevash's conjecture. Every $(t,r,n)$-Steiner system has chromatic index

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices, and let an $(n,q,r)$-Steiner system be a collection of $q$-sets containing every $r$-set exactly…

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices, and let an $(n,q,r)$-Steiner system be a collection of $q$-sets containing every $r$-set exactly…

Let $n>q>r\geq 1$ and let $X$ be an $n$-set. A design with parameters $(n,q,r,\lambda)$ is a set of $q$-subsets of $X$ in which every $r$-subset of $X$ belongs to exactly $\lambda$…

Existence conjecture. An $(n,r)$-pairs-triples design exists for each $n$ divisible by $6$ and odd $r$, $1 \leq r < n-1$, and for each $n \equiv 2$ or $4 \pmod{6}$ and…

Let $M$ be a matroid constructed from an affine or projective plane. Write $M_b$, $M_i$, and $M(i)$ for the matroids associated with the blocks and elements of the underlying Stein…

Let $X$ be an $n$-set, and let $S$ be a collection of $q$-subsets of $X$. A Steiner system with parameters $(n,q,r)$ is such that every $r$-subset of $X$ is contained in exactly on…

For integers $n\geq r\geq 2$, a fractional $(r-1,r,n)$-Steiner system in an $r$-uniform hypergraph $H$ is a function $\varphi:E(H)\to\mathbb{R}_{\geq0}$ such that…

Existence conjecture. A Steiner quadruple system of order $v$ with locality two and skip cost zero exists for all admissible parameters, namely all $v\geq 4$ satisfying $v\equiv 2$…

Kang–Kelly–Kühn–Methuku–Osthus conjecture. If

Existence Conjecture. For all $q$, $r$, and $\lambda$, and for sufficiently large $n$, these divisibility conditions are sufficient for the existence of an $(n,q,r,\lambda)$-design…

A Steiner system with parameters $(n,q,r)$ is a design in which every $r$-subset belongs to exactly one $q$-subset. For a partial $(n,q,r)$-Steiner system, the girth is the smalles…

A Steiner system with parameters $(n,q,r)$ is a design with parameters $(n,q,r,1)$. In a partial $(n,q,r)$-Steiner system, a $(j,i)$-configuration is a set of $i$ $q$-element subse…

Let $K(n,r)$ be the Kneser graph whose vertices are the $r$-subsets of $[n]$, with two vertices adjacent when they are disjoint. A subset $C$ of the vertices of a graph is a perfec…

Conjecture. There exists a $G$-invariant Steiner system on $\mega$.

A 1-rotational Steiner system S$(2,4,v)$ is a Steiner system with block size four admitting an automorphism consisting of one fixed point and a cycle of length $v-1$. On a point se…

Chromatic-number conjecture for derived STS. If