More results on overlapping squares

Three recent papers [5, 11, 8] have considered in complementary ways the combinatorial consequences of assuming that three squares overlap in a string. In this paper we provide a unifying framework for these results: we show that in 12 of 14 subcases that arise the postulated occurrence of three neighbouring squares forces a breakdown into highly periodic behaviour, thus essentially trivial and easily recognizable. In particular, we provide a proof of Subcase 4 for the first time, and we simplify and refine the previously established results for Subcases 11–14.