10 problems

Fractional chromatic number conjecture. The fractional chromatic number of every triangle-free cubic graph is at most

Let $m_1(\mathbb R^2)$ be the supremum of the upper densities of measurable sets $A\subseteq\mathbb R^2$ containing no two points at distance $1$. Erdős's conjecture. … The conject…

Let $d$ be sufficiently large, and let $G$ be a triangle-free graph. Write $\chi_f(G)$ for its fractional chromatic number, and say that $G$ is $d$-degenerate if every subgraph of…

Let $\chi_{f,0}(\mathbb{R}^2)$ denote the supremum of $\chi_f(G)$ over all finite unit distance graphs $G\subseteq\mathbb{R}^2$. Finitary fractional chromatic number conjecture. ……

Let $G$ be a finite unit distance graph in the Euclidean plane, and let $\alpha_1(G)$ denote its independence ratio. An independence ratio is attained at one quarter when…

Let $P$ be a $d$-regular edge-rooted pattern of depth $r$ and girth $2r+2$. For each edge-rooted pattern, let $\mathbf{e}(P)$ denote its constraint vector, and let…

Let $\rho(d,g)$ denote the maximum Hall ratio among graphs of maximum degree at most $d$ and girth at least $g$. Girth-shifting conjecture. The values presented in the table are up…

Let $G$ be a triangle-free graph with $m$ edges, and let $\chi_f(G)$ denote its fractional chromatic number. Edge-based fractional chromatic-number conjecture. As $m\to\infty$, eve…

Let $G$ be a triangle-free graph on $n$ vertices, and let $\chi_f(G)$ denote its fractional chromatic number. Fractional chromatic-number conjecture. As $n\to\infty$, every such gr…

Let $x\ge1$ be real and let $g\ge3$ be an integer. For a graph $G$, let $\chi_f(G)$ denote its fractional chromatic number, and let the girth of a graph be the length of its shorte…