You may of course use Szemeredi theorem, as suggested by Alexander Kalmynin.

If you need a more elementary argument, you may apply Van der Varden theorem as follows: assuming that the gaps are bounded by $T$, color every positive integer $n$ to the color $i\in \{1,\dots,T\}$ if $nT+i$ belongs to your set (so each large enough integer gets at least one color), and find a large monochromatic arithmetic progression. It corresonds to a large progression in the initial set.

You may of course use Szemeredi theorem, as suggested by Alexander Kalmynin.

If you need a more elementary argument, you may apply Van der Waerden theorem as follows: assuming that the gaps are bounded by $T$, color every positive integer $n$ to the color $i\in \{1,\dots,T\}$ if $nT+i$ belongs to your set (so each large enough integer gets at least one color), and find a large monochromatic arithmetic progression. It corresonds to a large progression in the initial set.