13 problems

Let $K^s_t$ be the complete $s$-uniform hypergraph on $t$ vertices, and let $\mathrm{sat}(n,K^s_t)$ denote its saturation number on $n$ vertices. Saturation-number conjecture. For…

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…

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 $k\geq 3$ and $1\leq \ell\leq k-1$. A $k$-uniform hypergraph is an $\e211$-hamiltonian saturated hypergraph if it is not $\e211$-hamiltonian, but adding any new edge creates a…

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…

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 $n$.

Pikhurko's conjecture. For every $r$-graph $F$, the limit

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…

Let $S_n(p,q)$ be the minimum number of edges in an $n\times n$ bipartite graph such that adding any new edge between its two classes creates a copy of $K_{p,q}$. Let…