243 problems

Determine whether there exists an absolute constant $c>0$ such that, for every integer $d\ge 1$, there is a constant $C_d>0$ for which every weakly $d$-degenerate ordered $3$-unifo…

For integers $n\ge k$ and $1\le t<k$, determine … where $\rho(\mathcal F)$ is the spectral radius of the adjacency tensor of the $k$-uniform hypergraph with edge set $\mathcal F$.…

For a finite triple system $S$, consider triple systems that omit $S$ and have uncountable chromatic number. (a) If one exists, must one exist with cardinality at most…

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.

Let a $(v,K,\lambda)$-pairwise balanced design be a $v$-vertex hypergraph whose $2$-degree is $\lambda$ and whose edge cardinalities belong to a set of integers $K$. A base block i…

For $r,e\geq3$, let $d_r(e)$ be the least integer $d$ such that every $r$-uniform hypergraph on $n$ vertices with no $e$ edges spanning at most $d$ vertices has $o(n^2)$ edges. Is…

For sets $A_1\subseteq A$, $B_1\subseteq B$, and $C_1\subseteq C$, a box $A_1\times B_1\times C_1$ is monochromatic under a coloring $f:A\to B\to C\to\operatorname{Fin}2$ if there…

For $k≥5$, let $F_k$ be the family of all three-uniform hypergraphs with $k$ vertices and $k-2$ edges. Is $ex_3(n,F_k)∼n^2/6$?

Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds. Any $r$-uniform hypergraph on $n$ vertices with at leas…

Let $f(n)$ be the minimum, over all $3$-uniform hypergraphs on $n$ vertices in which any two edges intersect in at most one vertex, of the maximum size of a vertex set containing n…

Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ the…

Let $m(n)$ be minimal such that there is an $n$-uniform hypergraph with $m(n)$ edges which is $3$-chromatic. Estimate $m(n)$.

Let $k\geq 2$ and $A_k\subseteq [0,1]$ be the set of $\alpha$ such that there exists some $\beta(\alpha)>\alpha$ with the property that, if $G_1,G_2,\ldots$ is a sequence of $k$-un…

Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic). Suppose a…

Does there exist an integer $k>2$ and a coloring of all $k$-element subsets of a $2k$-element set with $k+1$ colors such that, for every $(k+1)$-element subset $A$, the $k$-element…

Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree at least $7$?

Is there a constant $c>0$ such that every three-chromatic set system whose members all have size at least $t$ has an element belonging to at least $(1+c)^t$ members?

For fixed $r\geq3$ and all sufficiently large $k$, must every $r$-uniform hypergraph of chromatic number $k$ have at least $\binom{(r-1)(k-1)+1}{r}$ edges, with equality only for t…

Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges contains either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices wi…

Let $n,k,r,t$ be positive integers with $n≥kr+(t-1)(k-1)$. If the $r$-element subsets of an $n$-element set are coloured with $t$ colours, must some colour contain $k$ pairwise dis…

Does there exist a natural number $C$ such that, for all sufficiently large natural numbers $n$, there is a $3$-uniform hypergraph $H$ on the vertex set $\{0,\ldots,n-1\}$ having a…

How large must $\ell(n)$ be so that a uniformly random $3$-uniform hypergraph with $3n$ vertices and $\ell(n)$ edges contains $n$ pairwise vertex-disjoint edges with probability te…

Let $\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertic…

Let $\mathcal F$ be the family of all $3$-uniform hypergraphs with six vertices and three edges. Is $\operatorname{ex}_3(n,\mathcal F)=o(n^2)$?

Determine, for any $k>r>2$, the value of $\frac{\mathrm{ex}_r(n,K_k^r)}{\binom{n}{r}},$ where $\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$…