AC Arithmetic Combinatorics

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Section 16.6 Arithmetic Combinatorics

In recent years, a great deal of attention has been focused on topics in arithmetic combinatorics, with a number of deep and exciting discoveries in the offing. In some sense, this area is closely aligned with Ramsey theory and number theory, but recent work shows connections with real and complex analysis, as well. Furthermore, the roots of arithmetic combinatorics go back many years. In this section, we present a brief overview of this rich and rapidly changing area.

Recall that an increasing sequence \(a_1\lt a_2\lt a_3\lt \dots\lt a_t)\) of integers is called an arithmetic progression when there exists a positive integer \(d\) for which \(a_{i+1}-a_i=d\text{,}\) for all \(i=1,2,\dots,t-1\text{.}\) The integer \(t\) is called the length of the arithmetic progression.

Theorem 16.13.

For pair \(r,t\) of positive integers, there exists an integer \(n_0\text{,}\) so that if \(n\ge n_0\) and \(\phi:\{1,2,\dots,n\}\rightarrow\{1,2,\dots,r\}\) is any function, then there exists a \(t\)-term arithmetic progression \(1\le a_1\lt a_2\lt \dots\lt a_t\le n\) and an element \(\alpha\in\{1,2,\dots,r\}\) so that \(\phi(a_i)=\alpha\text{,}\) for each \(i=1,2,\dots,t\text{.}\)

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