The fine structure of threefold triple systems: υ ≡ 1 or 3 (mod 6)

The fine structure of a triple system of index λ is the vector (c1, … , cλ), where c1 is the number of triples appearing precisely i times in the system. We determine necessary conditions for a vector to be the fine structure of a threefold triple system and prove the sufficiency of these conditions for all υ ≡ 1, 3 (mod 6), υ ⩾ 19.