17 problems

Let $K_n^k$ be the complete $k$-uniform hypergraph on $n$ vertices, with its edges colored using two colors. A loose path is a hypergraph path in which consecutive edges intersect…

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of a $k$-uniform edge. A $k$-partite coloring of the complete graph $K_n$ is a coloring arising from a…

Let $p$, $k$ and $i$ be positive integers with $k\ge3$ and $1\le i<k$. For positive integers $n$, let $f_k(n,p,q)$ denote the minimum number of colors needed in an edge-coloring of…

Tightly connected hypergraph conjecture. For every $r\geq 3$, every $r$-coloring of $K_n^r$ contains a monochromatic tightly connected subgraph covering all vertices of $K_n^r$.

Rainbow tight Hamilton cycle conjecture. For every $k\geq 3$ and $\gamma>0$ there exist $c>0$ and $n_0>0$ such that if $(\mathcal{H},\phi)$ is an $n$-vertex colored $k$-graph with…

Fix integers $r,k\geq 2$, and let $F$ be a linear $k$-uniform hypergraph such that … Let $P$ be any pattern of $F$ on $t\geq 3$ classes, let $K_n^{(k)}$ be the complete $k$-uniform…

Let $k\geq 2$ and let $F$ be a linear $k$-uniform hypergraph such that … For an $n$-vertex $k$-uniform hypergraph $H$, let $c_{r,(F,P)}(H)$ denote the number of $(F,P)$-free $r$-co…

Let $r_3(n;2)$ be the minimum integer $N$ such that every $2$-coloring of the triples of an $N$-element set contains an $n$-element subset whose triples all have the same color. Er…

Monotonicity conjecture.

Weak 3-weighting conjecture. For each $r\geq 3$, every $r$-uniform hypergraph without isolated edges is weakly 3-weighted.

Strong weighting conjecture. For every $r\geq 3$, there is a constant $w=w(r)$ such that each nice $r$-uniform hypergraph is strongly $w$-weighted.

Dorbec–Gyárfás–Sárközy conjecture. Assume that $c\geq 2$, $2\leq t\leq r$, $c+t\leq r+1$, and $n$ is sufficiently large. Then every $c$-edge coloring of $K_n^r$ contains a monochro…

Gyárfás–Lehel–Sárközy–Szemerédi conjecture. For sufficiently large $n$, every $(r-1)$-edge coloring of $K_n^r$ contains a monochromatic Hamiltonian Berge-cycle.

Let $f(r,t)$ be the largest number of colors for which there exists an $(f,t)$-colored $r$-partite $r$-graph without a rainbow $t$-matching. Linear growth conjecture. For every $r$…

Let $r$, $s$, $t$, $k$ be positive integers. A $t$-colouring assigns one of $t$ colours to every edge of a complete $r$-uniform hypergraph, and an $s$-coloured matching of size $k$…

Let $k\geq 5$, and let $r$ and $\ell$ be positive integers satisfying … For a cover $C$ with elements $t_i$, let $H_{C,r}(n)$ denote the associated hypergraph, and let…

Let $G$ be a complete geometric graph with $n\geq 3$ vertices in general position. Let $i(V(G))$ denote the relevant parameter from the paper, and let a double transversal of plane…