A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST4. The result says that if A = 〈A0, A1, A2, A3〉 is a standard transitive and rich model of TST4, then A satisfies the 〈0, 0, n〉-property, for all n ≥ 2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in [5] to be equivalent to the consistency of NF. Such weak versions were introduced in [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of [5, Theorem 3.6], which is equivalent just to the 〈0, 0, 2〉-property.