Sets without $k$-term progressions can have many shorter progressions

TL;DR

This paper investigates the maximum number of shorter arithmetic progressions in sequences avoiding longer ones, revealing that the growth rate of such progressions is closely tied to Szemerédi's theorem.

Contribution

It establishes the asymptotic growth rate of the maximum number of s-term progressions in sequences without k-term progressions, answering a long-standing question of Erdős.

Findings

The limit of the logarithmic ratio of f_{s,k}(n) to n is 2.

Provides bounds linking the number of progressions to Szemerédi's theorem.

Shows the growth rate of progressions in progression-free sequences.

Abstract

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that $$\lim_{n \to \infty} \frac{\log f_{s,k}(n)}{\log n} = 2,$$ which answers an old question of Erd\H{o}s. In fact, we prove upper and lower bounds for $f_{s,k}(n)$ which show that its growth is closely related to the bounds in Szemer\'edi's theorem.