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Dec 10, 2019 at 1:37 comment added Fedor Petrov It looks so. You may simply find this pattern in $N$ lower rows where $N$ depends only on $c$. Also usual 2-dimensional van der Waerden is pretty enough: your set contains large grid of size about $N/2$ where you may find anything you need.
Dec 10, 2019 at 1:00 history asked Veridian Dynamics CC BY-SA 4.0