- every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and
- for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.

SOLVED

For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such that

- every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and
- for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.

Proved by Kwan, Sah, Sawhney, and Simkin [KSSS22b].

OPEN - $500

What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges.

A problem of Turán. Turán observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that
\[\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.\]
This is probably the truth. The current best upper bound is
\[\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},\]
due to Razborov [Ra10].

See also [712] for the general case.

OPEN

Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.

Prove that, for $r\geq 3$, \[\log_{r-1} R_r(n) \asymp_r n,\] where $\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like \[2^{2^{\cdots n}}\] where the tower of exponentials has height $r-1$?

OPEN

Let $F(n,\alpha)$ denote the largest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.

Prove that, for every $0\leq \alpha\leq 1/2$, \[F(n,\alpha)\sim c_\alpha\log n\] for some constant $c_\alpha$ depending only on $\alpha$.

It is easy to show that, for every $0\leq \alpha\leq 1/2$,
\[F(n,\alpha)\asymp_\alpha \log n.\]

Note that when $\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.

See also [161].

OPEN - $500

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.

Is there some constant $c>0$ such that \[R_3(n) \geq 2^{2^{cn}}?\]

OPEN

Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that
\[A\cup B= C\cup D\]
and
\[A\cap B=C\cap D=\emptyset.\]
Estimate $f(n;t)$ - in particular, is it true that for $t\geq 3$
\[f(n;t)=(1+o(1))\binom{n}{t-1}?\]

For $n=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1+o(1))n^{3/2}$.

Füredi proved that $f(n;3) \ll n^2$ and $f(n;3) > \binom{n}{2}$ for infinitely many $n$. More generally, Füredi proved that \[f(n;t) \ll \binom{n}{t-1}.\]

OPEN - $500

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 $n$ vertices so that there exists no $K_k^r$, a set of $k$ vertices which is covered by all $\binom{k}{r}$ possible $r$-edges.

Turán proved that, when $r=2$, this limit is
\[\frac{1}{2}\left(1-\frac{1}{k-1}\right).\]
Erdős [Er81] offered \$500 for the determination of this value for any fixed $k>r>2$, and \$1000 for 'clearing up the whole set of problems'.

See also [500] for the case $r=3$ and $k=4$.

SOLVED

Let $G$ be a $3$-uniform hypergraph with $6$ vertices and $3$ $3$-edges. Is it true that
\[\mathrm{ex}_3(n,G)=o(n^2)?\]

A conjecture of Brown, Erdős, and Sós. The answer is yes, proved by Ruzsa and Szemerédi [RuSz78] (this is known as the Ruzsa-Szemerédi problem).

In [Er81] Erdős asks whether the same is true for any $3$-uniform hypergraph on $k$ vertices with $k-3$ $3$-edges.

OPEN

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 $r+1$ vertices).

Is every $r$-hypergraph $G$ on $n$ vertices the union of at most $\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?

A conjecture of Erdős and Sauer.

SOLVED

How large should $\ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ vertices with $\ell(n)$ edges must contain $n$ vertex-disjoint edges?

Asked to Erdős by Shamir in 1979. This is often known as Shamir's problem. Erdős writes: 'Many of the problems on random hypergraphs can be settled by the same methods as used for ordinary graphs and usually one can guess the answer almost immediately. Here we have no idea of the answer.'

This is now essentially completely understood: Johansson, Kahn, and Vu [JKV08] proved that the threshold is $\ell(n)\asymp n\log n$. The precise asymptotic was given by Kahn [Ka23], proving that the threshold is $\sim n\log n$ (also for the general problem over $r$-uniform hypergraphs).

OPEN

Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)

SOLVED

Suppose $n\geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertices are $t$-coloured. Prove that some colour class must contain $k$ pairwise disjoint edges.

In other words, this problem asks to determine the chromatic number of the Kneser graph. This would be best possible: if $n=kr-1+(t-1)(k-1)$ then decomposing $[n]$ as one set $X_1$ of size $kr-1$ and $t-1$ sets $X_2,\ldots,X_{t}$ of size $k-1$, a colouring without $k$ pairwise disjoint edges is given colouring all subsets of $X_0$ in colour $1$ and assigning an edge with colour $2\leq i\leq t$ if $i$ is minimal such that $X_i$ intersects the edge.

When $k=2$ this was conjectured by Kneser and proved by Lovász [Lo78]. The general case was proved by Alon, Frankl, and Lovász [AFL86].

SOLVED

Let $r\geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-uniform hypergraph with chromatic number $k$ has at least
\[\binom{(r-1)(k-1)+1}{r}\]
edges, with equality only for the complete graph on $(r-1)(k-1)+1$ vertices?

When $r=2$ it is a classical fact that chromatic number $k$ implies at least $\binom{k}{2}$ edges. Erdős asked for $k$ to be large in this conjecture since he knew it to be false for $r=k=3$, as witnessed by the Steiner triples with $7$ vertices and $7$ edges.

This was disproved by Alon [Al85], who proved, for example, that there exists some absolute constant $C>0$ such that if $r\geq C$ and $k\geq Cr$ then there exists an $r$-uniform hypergraph with chromatic number $\geq k$ with at most \[\leq (7/8)^r\binom{(r-1)(k-1)+1}{r}\] many edges.

In general, Alon gave an upper bound for the minimal number of edges using Turán numbers. Using known bounds for Turán numbers then suffices to disprove this conjecture for all $r\geq 4$. The validity of this conjecture for $r=3$ remains open.

If $m(r,k)$ denotes the minimal number of edges of any $r$-uniform hypergraph with chromatic number $>k$ then Akolzin and Shabanov [AkSh16] have proved \[\frac{r}{\log r}k^r \ll m(r,k) \ll (r^3\log r) k^r,\] where the implied constants are absolute. Cherkashin and Petrov [ChPe20] have proved that, for fixed $r$, $m(r,k)/k^r$ converges to some limit as $k\to \infty$.

SOLVED

Does there exist an absolute constant $c>0$ such that, for all $r\geq 2$, in any $r$-uniform hypergraph with chromatic number $3$ there is a vertex contained in at least $(1+c)^r$ many edges?

In general, determine the largest integer $f(r)$ such that every $r$-uniform hypergraph with chromatic number $3$ has a vertex contained in at least $f(r)$ many edges. It is easy to see that $f(2)=2$ and $f(3)=3$. Erdős did not know the value of $f(4)$.

This was solved by Erdős and Lovász [ErLo75], who proved in particular that there is a vertex contained in at least \[\frac{2^{r-1}}{4r}\] many edges.

OPEN

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

A problem of Erdős and Lovász.

They do not specify what is meant by $3$-critical. One definition in the literature is: a hypergraph is $3$-critical if there is a set of $3$ vertices which intersects every edge, but no such set of size $2$, and yet for any edge $e$ there is a pair of vertices which intersects every edge except $e$. Raphael Steiner observes that a $3$-critical hypergraph in this sense has bounded size, so this problem would be a finite computation, and perhaps is not what they meant.

An alternative definition is that a hypergraph is $3$-critical if it has chromatic number $3$, but its chromatic number becomes $2$ after deleting any edge or vertex.

OPEN

Does there exist a $k>2$ such that the $k$-sized subsets of $\{1,\ldots,2k\}$ can be coloured with $k+1$ colours such that for every $A\subset \{1,\ldots,2k\}$ with $\lvert A\rvert=k+1$ all $k+1$ colours appear among the $k$-sized subsets of $A$?

A problem of Erdős and Rosenfeld. This is trivially possible for $k=2$. They were not sure about $k=6$.

This is equivalent to asking whether there exists $k>2$ such that the chromatic number of the Johnson graph $J(2k,k)$ is $k+1$ (it is always at least $k+1$ and at most $2k$). The chromatic numbers listed at this website show that this is false for $3\leq k\leq 8$.

OPEN

Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$, such that any two edges have non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\gg r$ many vertices?

A problem of Erdős and Shelah. The Fano geometry gives an example where there are no two edges which meet in $r-1$ vertices. Are there any other examples?

Erdős and Lovász [ErLo75] proved that there must be two edges which meet in $\gg \frac{r}{\log r}$ many vertices.

OPEN

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 $k$-uniform hypergraphs with
\[\liminf \frac{e(G_n)}{\binom{\lvert G_n\rvert}{k}} >\alpha\]
then there exist subgraphs $H_n\subseteq G_n$ such that $\lvert H_n\rvert \to \infty$ and
\[\liminf \frac{e(H_n)}{\binom{\lvert H_n\rvert}{k}} >\beta,\]
and further that this property does not necessarily hold if $>\alpha$ is replaced by $\geq \alpha$.

What is $A_3$?

A problem of Erdős and Simonovits. It is known that
\[A_2 = \left\{ 1-\frac{1}{k} : k\geq 1\right\}.\]