14 problems

Let $R=\{(\alpha,\beta,\gamma)\in[0,1]^3:\alpha\beta+\gamma>1,\alpha\gamma+\beta>1,\beta\gamma+\alpha>1\}$, and let … Define…

The density domination exponent conjecture.

Degenerate-graph density conjecture. For every fixed $k$-degenerate graph $H$,

Small-density profile conjecture. For every graph $G_l$,

Nagy's K-or-SK type conjecture. Every graph is either of type K or of type SK.

Monotonicity conjecture. For each graph $G_l$, the function $q(G_l,\beta)$ is increasing in the edge density $\beta$.

Three-step extremal profile conjecture. For every graph $G_l$ and every $\beta\in[0,1]$,

Let $d_{C_5}(p)$ denote the minimum possible density of copies of the 5-cycle $C_5$ in graphs with edge-density $p$. For an integer $k\geqslant 2$ and a real number…

Let $g_3(\lambda)$ denote the asymptotic minimum triangle density among graphs of edge density $\lambda$. Nordhaus–Stewart's conjecture. … The conjecture proposes a convex lower bo…

Let $H$ be a graph with edge densities $b3_e$. For each proper labeling $f$, consider the weighted monotone-path tree $T_f(H)$, with the edge weights interpreted as densities, and…

Let $G$ be a graph, let $K_r$ be the complete graph on $r$ vertices, and let $\overline{K}_r$ be its complement. Write $d(H;G)$ for the normalized density of copies of a graph $H$…

A triangle-free graph is a graph containing no 3-cycles; a pentagon is a cycle of length $5$. The density of pentagons is the number of pentagons divided by the appropriate normali…

Skew blow-up conjecture. There is a constant $c>0$ such that

Balanced blow-up conjecture. For every $t\geq 2$,