18 problems

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…

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…

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 $K_{s,t}$ 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…

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 $P$ 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…

Let $N$ and $n$ be positive integers, and let $\mathcal{F}\subset \mathcal{P}(N)\times\mathcal{P}(N)$ have size … Assume that every $(A_j,B_j)\in\mathcal{F}$ satisfies…

Tree-poset consecutive-level minimizer conjecture. There exists a constant $c=c(T)$ with $0<c\le1$ such that, if $E\le c\binom{n}{\lfloor n/2\rfloor}$, then some family…

Butterfly supersaturation conjecture. Let $E=E(n)\le K(n,\lceil n/2\rceil+1)$. If $n$ is large enough, then the minimum number of butterflies a family $\mathcal F\subset 2^{[n]}$ o…