Incompleteness in Finite Combinatorics

There exist recursive functions over \(\mathbb {N}\) that grow so quickly that PA cannot prove their totality. In 1977, Paris and Harrington introduced such a function \(\mathrm {PH}(n, k, l)\) , which can be represented in PA but where \((*) :=\) “ \(\forall n,k,l \: \exists ! m \: \mathrm {PH}(m, n, k, l)\) ” is true over \(\mathbb {N}\) but not provable in PA.