Horn sentences excluding a prime (Q1080848)

A first order sentence \(\sigma\), in the language whose unique relation symbol is equality, is said to exclude a prime p if and only if the models of \(\sigma\) are all the sets of cardinality different from p. In this paper the author proves that for any prime p there exists a Horn sentence excluding p whose size is \(O(p^ 3 \log p)\). This result improves a previous solution of size \(O(p^ 5 \log p)\) by the same author. K. I. Appel obtained the first solution, but it was superexponential.