Minimum-size minimal integer exclusion sets of length three

Determine the minimum number of positive tuples in a minimal integer exclusion set of length three, classify all sets attaining this minimum, and prove that exactly nine attain it, with the specified exclusion set \(\overline{O}\) being the only reversible one.

Background

The paper classifies all minimal exclusion sets of length two over the integers, obtaining eighteen sets when reversals are included, and constructs an infinite exclusion set that is strictly stronger than every finite exclusion set. It then discusses the substantially more complicated length-three case, reporting experimental evidence that there may be hundreds of minimal exclusion sets, each containing at least thirteen positive tuples.

The authors formulate a conjecture about the sharp lower bound of thirteen positive tuples, the number of extremal sets, and reversibility. The conjecture is unresolved in the paper and is therefore included as an explicit open problem.