Applications of the Bergelson–Leibman and the Mordell–Faltings Theorems

To achieve a girth 12 unit-distance graph, Paul O’Donnell alters the set D of allowable constant differences. This changes which sets are in S (i.e., which sets of the foundation vertices get odd cycles attached). It is no longer enough for the sets in S to have intersection of size at most one, as we required in Chap. 48 . In addition, O’Donnell requires now that no three sets in S intersect pairwise. How does one achieve this?