13 problems

Let $r\geq 3$ be a fixed integer. For a path $P_k$ with $k$ edges, let $\mathcal{B}(P_k)$ denote its Berge hypergraph, and let $\mathrm{schex}_r(n,\mathcal{B}(P_k))$ be the strong…

For a 3-uniform hypergraph, let $\mathrm{sat}_3(n,\text{Berge-}K_4)$ denote the minimum number of hyperedges in a Berge-$K_4$-saturated hypergraph on $n$ vertices. Here a hypergrap…

Mubayi–Zhao conjecture. Given positive integers $n,r,s,t$ with $2\leqslant t<\min\{r,s\}$, there exists $n_0>0$ such that for $n>n_0$,

Let $t>r\geq2$ and suppose $(r-1)^r\mid n-r$. Let $H^*$ be a linear $r$-graph attaining the spectral extremal value for Berge-$K_{3,t}$-free linear $r$-graphs, and let…

Let $F$ be a graph, and let $sat_k(n,\mathrm{Berge}\text{-}F)$ denote the saturation number for $k$-uniform hypergraphs on $n$ vertices avoiding Berge-$F$. Linear saturation conjec…

Let $F$ be a graph, let $k$ be a positive integer, and let $\mathrm{sat}_k(n,\text{Berge-}F)$ denote the least number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on…

Let $F$ be a $u$-uniform hypergraph, and let a Berge copy of $F$ be a hypergraph $H$ for which $V(F)\subset V(H)$ and there is a bijection $f:E(F)\to E(H)$ such that…

Let $F$ be a graph, and let a Berge copy of $F$ be a hypergraph $H$ for which $V(F)\subset V(H)$ and there is a bijection $f:E(F)\to E(H)$ such that $e\subset f(e)$ for every…

Let $R_r(\mathcal B K_n;c)$ be the smallest $N$ such that every $c$-coloring of the $r$-edges of the complete $r$-uniform hypergraph $K_N^r$ contains a monochromatic Berge copy of…

Higher-uniformity Berge saturation conjecture. For every $3\leq r<k$ and every $r$-uniform hypergraph $F^{(r)}$,

Linearity conjecture. For every uniformity $k$ and every graph $F$, the saturation number $\operatorname{sat}_k(n,\text{Berge-}F)$ grows linearly in $n$.

Let $k$ be fixed, and let $\mathcal{F}$ be any fixed finite family of graphs. Write $\operatorname{sat}_k(n,\text{Berge-}\mathcal{F})$ for the minimum number of edges in a $k$-unif…

Let $\operatorname{sat}_k(n,\text{Berge-}P_m)$ denote the minimum number of edges in a $k$-uniform hypergraph on $n$ vertices that is Berge-$P_m$-saturated, and let $a^{(k)}_m$ be…