New results in classical and arithmetic Ramsey theory

Abstract: For $r,k\in \N$, Ramsey's Theorem says that there exists a least positive integer $R_r(k)$ such that every $r$-coloring of the edges of a complete graph on $N\geq R_r(k)$ vertices yields a monochromatic complete subgraph on $k$ vertices. This fact can be applied to deduce Schur's Theorem, which says that there exists a least positive integer $S_r(k)$ such that every $r$-coloring of $\{1,2,\dots,N\}$ for $N\geq S_r(k)$ yields a monochromatic solution to the equation $x_1+x_2+\cdots+x_{k-1}=x_k$. Here we discuss new findings related to these two classical results. First, we derive explicit upper bounds on $R_r(k)$, established through the pigeonhole principle and careful bookkeeping, that improve upon previously documented bounds. Second, we present an extension of Schur's Theorem to higher-dimensional integer lattices, with the additional restriction that the vectors on the left hand side of the equation are linearly independent.

This includes joint work with six (at the time) Millsaps College undergraduate students: Vishal Balaji, Powers Lamb, Andrew Lott, Dhruv Patel, Sakshi Singh, and Christine Rose Ward.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)