6 problems

Let $G$ and $H$ be graphs. A pair $(G,H)$ is Ramsey-infinite if it has infinitely many Ramsey-minimal graphs. An odd star is a star with an odd number of leaves, and a $K_2$-compon…

Let $M_a^\mathrm{nest}$ be a nested matching with even parameter $a$, and let $S_b$ be a start-central star with parameter $b$. Nested-matching–star conjecture. For any even…

Let $S_a$ be the start-central star with parameter $a$, and let $P_b^\mathrm{alt}$ be an alternating path of order $b$. Star–alternating-path conjecture. For any $a \ge 3$ and…

Let $S_a$ be the start-central star with parameter $a$, and let $P_b^\mathrm{mon}$ be a monotone path of order $b$. Star–monotone-path conjecture. For any $a \ge 3$ and $b \ge 4$,…

Let $S_k$ be the star with $k$ vertices, and let $K_r$ be a clique with $r>2$. For $n$ not divisible by $k-1$, the Gan–Loh–Sudakov conjecture. The extremal graph for…

Let $S_3=K_{1,3}$ be the star with three edges, let $S_3^+$ be its $3$-uniform expansion, and let $\operatorname{ex}_3(n,S_3^+)$ denote the maximum number of edges in an $n$-vertex…