4 problems

Determine the exact value of the Ramsey number $R(4,20)$, defined by…

Let $P_{\text{ASEP}}^{n}$ be the asymmetric subtour elimination polytope, let $\boldsymbol{x}$ be a vertex of this polytope, and let $\mathcal{N}(\boldsymbol{x})$ denote its neighb…

Let $G=\mathbb{Z}_8^2$ and let a subset $S\subset G$ be primitive if it is not contained in a proper coset of a subgroup of $G$ and generates $G$. A subset is formally dual if ther…

Let $k\geq 10$ be an integer, and let $p$ and $q$ be integers satisfying $p\geq k+2$ and $q\geq k+1$. Write $r_k(p,q)$ for the smallest integer $n$ such that every red-blue colorin…