33 problems

Let $[n]$ be the ground set, and let a $k$-chain be a sequence of $k$ distinct subsets of $[n]$ ordered by strict inclusion. For each family size $a$, consider the centralized fami…

Mubayi's conjecture. If $\mathcal F$ is a stable non-$r$-partite $r$-graph, then, for every positive integer $q$ and all sufficiently large $n$, every $n$-vertex $r$-graph with…

Let $G$ be an $n$-vertex graph with $m$ edges. A book of size $k$ is a graph consisting of $k$ triangles sharing a common edge, and $\operatorname{bk}(G)$ denotes the largest such…

Bollobás–Leader conjecture. For every $s$, there is some $1\leq \ell\leq k$ such that an $ℓ$-ball minimizes the number of disjoint pairs among all $k$-uniform families of size $s$.

Let $k$ and $n$ be positive integers with $2\leq k\leq n+1$, and let $\mathcal{A}\subseteq 2^{[n]}$ be a family of size $m$ greater than Erdős' bound for families containing no $k$…

O'Neill's supersaturation conjecture. If $\mathcal{A}\subset 2^{[n]}$ consists of even-sized subsets and

Let $F$ be a color-critical graph with order $f=|F|$ and chromatic number $\chi(F)=r+1\geqslant4$. Write $N_F(G)$ for the number of copies of $F$ in an $m$-edge graph $G$, and let…

Let $F_2$ be the bowtie graph, let $\tau(G)$ denote the number of copies of $F_2$ in a graph $G$, and let $K_{s,t}^{q}$ be obtained from $K_{s,t}$ by adding $q$ pairwise disjoint e…

For an integer $k\geq 1$, define … Let $F_k$ be the friendship graph and let $\tau_k(G)$ denote the number of copies of $F_k$ in $G$. Let $K_{s,t}^{q}$ be obtained from $K_{s,t}$ b…

Let $F_2$ be the bowtie graph, let $\tau(G)$ denote the number of copies of $F_2$ in a graph $G$, and let $K_{s,t}^{q}$ be obtained from $K_{s,t}$ by adding $q$ pairwise disjoint e…

Let $G$ be an $n$-vertex graph, and let $q(G)$ denote its signless Laplacian spectral radius. For fixed $k\geq 3$ and sufficiently large $n$, consider graphs whose signless Laplaci…

For an $r$-partite $r$-graph $F$, let $s(F)$ be its Sidorenko gap, defined by … Assume $r\geq 3$ and … for some $0<\alpha<r-1$. The hypergraph supersaturation conjecture. There exi…

Let $F$ be a bipartite graph such that … for some $0<\alpha<1$. Let $G$ be an $n$-vertex graph, and write $v(F)$ and $e(F)$ for the numbers of vertices and edges of $F$. Simonovits…

Let $K_{3,t}$ be the complete bipartite graph with parts of sizes $3$ and $t$, and let $K_3$ denote the triangle. A graph $F$ is $K_3$-Erdős–Simonovits when it satisfies the genera…

Let $F$ be a bipartite graph. Consider graphs with a prescribed number of edges, and count copies of $F$ in those graphs. Erdős–Simonovits–Sidorenko conjecture. Among all graphs wi…

Erdős–Katona conjecture. The family $\mathcal{F}$ contains at least

Let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$, let $K_r$ be the complete graph on $r$ vertices, and let $\mathcal{N}(F,G)$ denote the number of copi…

Let $K_{2,t}$ denote the complete bipartite graph with parts of sizes $2$ and $t$, and let $\mathcal{N}(F,G)$ denote the number of copies of $F$ in $G$. The $K_{2,t}$ clique supers…

For integers $n,k\geq 1$, let … where $t(G)$ is the number of triangles in $G$ and $\operatorname{ex}(n,C_3)$ is the Turán function of $C_3$. Erdős's conjecture. … for all…

Odd-town supersaturation conjecture. Let $n\geq 1$ and fix $1\leq s\leq n$. If $[?]$ is a collection of odd-sized subsets of an $n$-element set with $|[?]|\geq n+s$, then

Supersaturation conjecture. Let $n\geq 1$ and fix $3\leq s\leq 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}$. If $[?]\subset 2^{[n]}$ consists of even-sized subsets with…

Let $k$ and $\ell$ be positive integers with $k>\ell$. For an $n$-vertex graph containing at least $m$ copies of the path $P_k$, let $\operatorname{satex}(n,P_k:m,P_{\ell})$ denote…

Let $k>1$, let $q>1$ be fixed, and set $t=q(k-1)+1$. For an $n$-vertex graph containing at least $m$ copies of the path $P_k$, let $\operatorname{satex}(n,P_k:m,P_t)$ be the minimu…

Let $P$ be a poset and $\varepsilon>0$. Let $La(n,P)$ and $La^*(n,P)$ be the largest sizes of, respectively, $P$-free and induced $P$-free families in $2^{[n]}$. Let $e(P)$ and…

Let $r\geq s\geq 3$, let $\epsilon>0$, and let $G$ be a graph with $m=e(G)$ edges. Write $\operatorname{k}_s(G)$ for the number of copies of $K_s$ in $G$. Supersaturation conjectur…