Roth’s function is defined as the largest size of a subset of without arithmetic progressions. We may “force” the arithmetic progressions into the measure, and instead of taking the usual measure we may consider where denotes the set of all AP’s in and equals if , otherwise, and the are constants. The advantage of doing this is that we now have an optimization problem on all subsets of instead of the subsets without APs.

For example, we may take all the ‘s equal to , but the extremal subsets (those which maximize ) do not appear to exhibit a very interesting structure. Perhaps there are better choices for the constants.

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