23 problems

Extremal characterization conjecture. $f_r(s)$ achieves the maximum for either $\mathcal{K}_{s+r-1}^r$, or the hypergraph $\mathcal{G}$ on $s+r$ vertices obtained by this minimum-e…

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…

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…

A $(j,i)$-configuration in a family of sets is a collection of $i$ members spanning at most $j$ ground-set elements; the girth of a triangle packing is the smallest $i\geq 2$ for w…

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…

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 $M$ 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…

Chromatic-number conjecture for derived STS. If

Resolvable-derived-STS conjecture. If

For integers $q>r\ge2$, an $(n,q,r)$-Steiner system is a collection of $q$-element subsets of an $n$-element set in which every $r$-element subset lies in exactly one block. A…

Given $n\ge q>r\ge2$, an $(n,q,r)$-Steiner system is a collection of $q$-element subsets of an $n$-element set in which every $r$-element subset lies in exactly one block. A…

A Steiner triple system ${\rm STS}(v)$ is a $2$-$(v,3,1)$ design. A zero-sum $n$-flow of a design $\mathcal{D}=(X,\mathcal{B})$ is a map … such that the sum of the values over all…

Affine-plane representability problem. Determine whether

For an integer $e\ge 2$, an $(n,3,2)$-design is an $e$-sparse Steiner triple system if it is simultaneously $\mathscr{G}_3(i+2,i)$-free for every $2\le i\le e$. Erdős's conjecture.…

Let $m\ge 3$ be an odd integer, let $\alpha$ be a primitive element of $\operatorname{GF}(3^m)$, and let $\mathcal C_1$ and $\mathcal C_2$ be the ternary linear codes defined above…

Let $p(n,r)$ denote the number of paving matroids on an $n$-element ground set of rank $r$, and let $p_k(n,r)$ denote the number of paving matroids without hyperplanes of cardinali…

Let $\ell \ge 4$. Let $m$ be the total number of steps in the high-girth triple-process that produces a partial Steiner system with girth greater than $\ell$. High-girth process co…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, let $\mathcal{H}_k(n;p)$ be the random $k$-uniform hypergraph obtained by retaining each edge independently with…

Weight-support system conjecture. All codewords of weight $3$ of a perfect code containing a Preparata code, and all codewords of weight $4$ of an extended perfect code containing…

Let $n$ be a prime satisfying … A coset complete 3-dimensional subspace is a 3-dimensional subspace of $\mathbb{F}_2^n$ used as a generator for the construction described in the pa…