10 problems

Let $M_a^\mathrm{nest}$ be a nested matching with even parameter $a$, and let $P_b^\mathrm{mon}$ be a monotone path of order $b$. Nested-matching–monotone-path conjecture. For any…

Let $M_a^\mathrm{nest}$ and $M_b^\mathrm{nest}$ denote nested matchings with parameters $a$ and $b$. Nested-matching conjecture. Let $a, b \ge 4$ be divisible by four. Then … A con…

Let $K_a$ be a complete graph of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Complete-graph cyclic conjecture. For any $a, b \in \mathbb{N}$, we have … The c…

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 $C_a^\mathrm{mon}$ be a monotone cycle of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Cyclic monotone-cycle–alternating-path conjecture. For any…

Let $P_a^\mathrm{alt}$ be an alternating path of order $a$ and $P_b^\mathrm{ralt}$ a reverse alternating path of order $b$. Alternating–reverse-alternating conjecture. For any…

Let $P_a^\mathrm{alt}$ be an alternating path of order $a$ and $P_b^\mathrm{mon}$ a monotone path of order $b$. Mixed path conjecture. For any $a \ge 3$ and $b \ge 4$, we have … Th…

Let $P_a^\mathrm{mon}$ and $P_b^\mathrm{mon}$ denote monotone paths of orders $a$ and $b$, respectively, and let $R_\mathrm{cyc}$ be the cyclic Ramsey number. Cyclic monotone-path…

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring, and let $J_n$ denote the ideal associated with the cyclic graph on $n$ vertices as in the paper. The Stanley depth conjecture. For a…