On nearly consecutive sequences without long arithmetic progressions

Abstract

A sequence $a_1,\ldots,a_n$ of integers is nearly consecutive if $a_{i+1}-a_i \in \{1,2\}$ for $1 \leq i \leq n-1$. We prove that there are nearly consecutive sequences of length $Ω(2^k/k^2)$ that contain no $k$-term arithmetic progression. This improves on the previous best known bound of Alon and Zaks from 1998. We also prove a generalization for sequences with bounded gaps.