On a combinatorial principle of Hajnal and Komjáth

In their paper [3], Hajnal and Komjáth define the following combinatorial principle:Definition 1.1. Suppose κ is an infinite cardinal and n < ω. Then Hn(κ) is the statement: There is a function F: [κ]n → [[κ]ω]≤ω such that(a) ∀A ∈[κ]n ∀Y ∈ F(A)(Y ⊆ min (A)), and(b) .Hn(κ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4].In [3] Hajnal and Komjáth prove the consistency of ZFC + GCH + ∀n ∈ ω(Hn + 1(ωn + 1)), relative to an ω-Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H1 (ω1) is true, and Laver proved that V = L implies H2(ω2). In this paper we go one step further and prove V = L → H3(ω3). Unfortunately our methods do not appear to give Hn (ωn) for n ≥ 4.Most of our notation is standard. If X is any set and κ is a cardinal number then [X]κ is the set of subsets of X with cardinality κ, and [X]≤κ is the set of subsets of X with cardinality ≤ κ. If X is a set of ordinals then tp(X) is the order type of X.