63 problems

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

Let ${\cal C}_{\geq k}$ denote the family of cycles of length at least $k$. For $2\leq a\leq\lfloor (k-1)/2\rfloor$, let $H(n,a,k)$ be a graph with vertex set $A\cup B\cup C$ such…

Kang–Kim–Liu conjecture. Under this hypothesis,

Ghosh–Győri–Martin–Paulos–Xiao conjecture. For each $k\geq 7$ and all sufficiently large $n$,

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

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…

For an integer $\ell\geq 1$, let $C^{\square}_{\ell}$ be the prism graph consisting of two vertex-disjoint $\ell$-cycles together with a perfect matching between corresponding vert…

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

Asymptotic planar Turán conjectures.

Erdős–Sárközy–Sós conjecture. There is a constant $c$ such that (a) $ex(m,n,C_6)<c(mn)^{\frac{2}{3}}$ when $n>m\geq n^{\frac{1}{2}}$, and (b) $ex(m,n,C_6)<2n+c(mn)^{\frac{2}{3}}$ w…

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…

A graph $H$ is $r$-degenerate if each of its subgraphs has minimum degree at most $r$. Let $H$ be a bipartite $r$-degenerate graph. Erdős's conjecture. … The conjecture is known wh…

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…

Spiro's conjecture. For every graph $F$, there are infinitely many realizable exponents for $F$.

Erdős–Simonovits Rational Exponents Conjecture. Every rational number $r\in[1,2]$ is a realizable exponent.

McKinley–Spiro conjecture. Almost surely,

Erdős–Füredi–Gould–Gunderson conjecture. The same formula should hold for all $1\leq k\leq n/4$. Their conjecture extends the known range of the exact Turán-number formula for fans…

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 $H$ 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 $m,n$ be positive integers, and let $ex(m,n,\mathcal{F})$ denote the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that contains no member of the…

Li's conjecture. For every relevant $n$,

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 $F$ be any graph, and let $\mathrm{ex}(n,F)$ denote the maximum number of edges in an $F$-free graph with $n$ vertices. Power-law conjecture. There exists a constant $\alpha$ s…