Ramsey Theory for Diffsequences

Abstract: Van der Waerden's theorem states that any coloring of $\mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r,k)$ which is the smallest $n$ such that any $r$-coloring of $\{1,2,\cdots,n\}$ guarantees the presence of a monochromatic arithmetic progression of length $k$.

It is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion, introduced by Landman and Robertson, is that of a $D$-diffsequence, which is an increasing sequence $a_1 < a_2 <\cdots< a_k$ in which the consecutive differences $a_i-a_{i-1}$ all lie in some given set $D$. For each $D$ that exhibits a van der Waerden-type result, we let $\Delta(D,k;r)$ represent the analogue of the van der Waerden number $W(r,k)$. One question of interest is to determine the possible behaviors of $\Delta$ as a function of $k$. In this talk, we will demonstrate that it is possible for $\Delta(D,k;r)$ to grow faster than polynomial in $k$. Time permitting, we will also discuss a class of $D$'s for which no van der Waerden-type result is possible.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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