On nearly consecutive sequences without long arithmetic progressions

A sequence $a_1,\dots ,a_n$ of integers is nearly consecutive if $a_{i+1}-a_i\in \{1,2\}$ for $1\le i\le n-1$. We prove that there are nearly consecutive sequences of length $\mathrm{\Omega }(2^k\mathrm{/}{k}^2)$ 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.