9 problems

Disjoint-union conjecture. For every two graphs $G$ and $H$,

Let $W_5$ be the join of the cycle $C_5$ with $K_1$, let $\alpha(G)$ denote the independence number of $G$, and let $W_5^{\Box 4}$ be the fourth Cartesian power of $W_5$. Fourth-po…

Let $W_t$ be the join of the cycle $C_t$ with $K_1$, and let $d4b0(G)$ denote the ultimate independence ratio of a graph $G$. For odd wheels, $d4b0(W_{2t+1})$ is bounded below by…

Let $\delta_{\mathrm{Croft}}$ be Croft's density bound for measurable 1-avoiding subsets of the plane. Let $m_1(\mathbb{R}^2)$ be the measurable independence ratio of the plane, an…

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 $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k\geq8$ with…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k>6$ with…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k,\ell\geq 1$. The i…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Fix an odd integer…