12 problems

Burr–Erdős conjecture. Every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$.

Goldberg's density conjecture. For any graph $G$, if $\Gamma(G)\le\Delta(G)-1$, then

Let $s$ and $t$ be integers with $3\leq s\leq t-2$. An extremal graph for the generalized Ramsey–Turán density $\varrho_s(K_t)$ is a graph admitting either an $(s,t-1-s)$-partition…

Kaufmann–Ueckerdt's density conjecture. The density of simple fan-planar topological graphs with non-homeomorphic parallel edges and non-trivial loops remains

For a graph $G$, let $m(G)=\max_{G'\subseteq G,\,v(G')\geq 1}e(G')/v(G')$ denote its maximum average density, and let $m_2(G)$ denote its maximum $2$-density. Forest deletion conje…

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

Erdős–Hajnal–Simonovits–Sós–Szemerédi conjecture. The asymptotic extremal graphs $G$ for $\varrho_p(q)$ have a partition

Let $G$ be a graph. A graph is Ramsey-cofinite when it has the cofinite monochromatic embedding property defined in the paper, and it is Ramsey-lower-dense when it has positive Ram…

For each $d\in\mathbb{N}$, let $H_d$ be the graph defined in the source's Proposition referenced by prop:constr, and let $\operatorname{\overline{Rd}}(H_d)$ denote its two-color Ra…

Conlon–Fox–Sudakov density conjecture. For every graph $H$ there exist $\epsilon,\sigma>0$ such that for every $H$-free graph $G$ on $n>1$ vertices and every $c$ with…

For each integer $k\ge 2$, let $d_k=d_k(K_k)$ denote the critical edge density of the complete graph $K_k$. Pfender's recurrence conjecture. The critical edge densities satisfy … T…

The density conjecture for $5/2$-critical graphs. If $G$ is a $5/2$-critical graph, then