37 problems

For each integer $k\geq2$, consider the Cartesian product $C_{2k}\square P_2$ of the cycle $C_{2k}$ and the path $P_2$. The notation $\operatorname{ex}(n,H)$ denotes the maximum nu…

Let $ex_{\mathcal{P}}(n,2C_4)$ denote the maximum number of edges in an $n$-vertex planar graph containing no two vertex-disjoint copies of the cycle $C_4$. Planar Turán conjecture…

For graphs $F$ and $G$, their Cartesian product $F\Box G$ has vertex set $V(F)\times V(G)$, with $(u,v)$ adjacent to $(u',v')$ if and only if either $u=u'$ and $vv'\in E(G)$, or…

Bukh–Conlon conjecture. For any balanced rooted tree $(T,R)$ and any natural number $\ell$, we have

Let $k_0\geq 1$ be an integer. For a graph $H$, write $\operatorname{ex}(n,H)$ for its Turán number, and call a rational number $r\in[1,2]$ realizable as a Turán exponent if there…

Generalized Turán number exponent conjecture. There exists $t_0$ such that, for all $t\geq t_0$,

For a graph $H$, let $\mathrm{ex}(n,H)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $H$. Erdős–Simonovits rational exponents conjecture. For eve…

Let $TP_4$ denote the triangular pyramid graph with four levels, and let $\operatorname{ex}(n,TP_4)$ be the maximum number of edges in an $n$-vertex graph containing no copy of…

Let $k_1\geq k_2\geq\ldots\geq k_m\geq 3$ with $k_1>3$, and let … be the disjoint union of paths. For integers $n\geq m\geq 3$, write $c(n,a,b)$ and $c(n,a)$ as in the preceding de…

Let $n\geq k\geq 4$. For a graph $F$, let $\widehat{F}$ denote the graph obtained by adding a vertex adjacent to every vertex of $F$, and let $\operatorname{ex}(n,H,F)$ be the maxi…

Let $H$ be a graph, with $v(H)$ vertices, $e(H)$ edges, and maximum degree $\Delta(H)$. A rational number is realizable for $H$ if it occurs as an exponent in the generalized Turán…

Let $T$ be a tree whose smaller color class has size $k$, and let $\widehat{T}$ denote the suspension of $T$. Let $f(n,k)$ be the bound defined earlier in the paper. Conjecture on…

Let $r\ge 3$, and let $\mathcal{F}$ be a degenerate finite family of $r$-graphs satisfying … for some constant $\alpha>r-2$. Then there exist constants $\beta>0$, $c>0$, and $C>0$…

Let $ex_{\mathcal{P}}(n,H)$ denote the maximum number of edges in a planar graph with $n$ vertices that does not contain $H$ as a subgraph. For $k\geq 4$, let $\Theta_k$ denote the…

Let $C_k$ be the cycle on $k$ vertices, and let $\operatorname{ex}_{\mathcal{P}}(n,C_k)$ denote the maximum number of edges in an $n$-vertex planar graph containing no copy of…

Ferber–McKinley–Samotij's conjecture. There exists an $\varepsilon>0$ such that

For positive integers $d$ and $t$, let $F_t^{(d)}$ be the $d$-dimensional grid with vertex set $[t]^d$, where two vertices are adjacent when they differ by exactly one in exactly o…

Let $T$ and $S$ be trees, each with at least one edge, and let $T\Box S$ denote their Cartesian product. For a positive integer $n$, let $\mathrm{ex}(n,T\Box S)$ be the maximum num…

Let $C_8$ be the cycle on eight vertices, and let $\operatorname{ex}(n,H)$ denote the maximum number of edges in an $H$-free graph on $n$ vertices. $C_8$ extremal-number conjecture…

Let $TP_4$ denote the triangular pyramid of $4$ layers, and let $\operatorname{ex}(n,TP_4)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $TP_4$. Asym…

Let $TP_3$ denote the triangular pyramid of $3$ layers, and let $\operatorname{ex}(n,TP_3)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $TP_3$. Turá…

Existence of Bukh–Conlon densities in every fractional class. For every $s,a\in\mathbb{N}$ with $s<a$, there exists $m\in\mathbb{N}^+$ such that $m+s/a$ is a Bukh–Conlon density.

Let $r\in(1,2)$ be rational. For a graph $F$, write $\operatorname{ex}(n,F)$ for the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Realizability of rati…

Grzesik–Janzer–Nagy conjecture for even cycles. For every pair of integers $r,k\geq2$,

Grzesik–Janzer–Nagy conjecture. If