Friedman's finite trees

Define \([n]^k\) as all the \(k\)-element subsets of \(\{0, 1, 2, \ldots, n-1\}\).

For \(k,n \geq 1\), let \(\text{TR}(k,n)\) be the set of all \(k\)-trees with children in \([n]^k\).

An insertion domain in \(\text{TR}(k,n)\) is a nonempty set \(S \subseteq \text{TR}(k,n)\) such that for all \(T \in S\), \(x \in [n]^k\), \(x\) dominating \(T\), there exists \(T' \in S\) such that \(T \subseteq T'\) and \(\text{Ch}(T') = \text{Ch}(T) \cup \{x\}\). An insertion domain in \(\text{TR}(k,n)\) is initial iff it includes the trivial \(k\)-tree.

Let \(F(k, p)\) be the least \(n\) such that every nonincreasing initial insertion domain in \(\text{TR}(k, n)\) contains a \(k,n\)-tree in which all \(k\)-element subsets of some \(p\)-element set are vertices with the same entirely lower ancestors.