2- and 3-existentially closed tournaments

Abstract

A tournament has property $P_{k}\,  (k\geq 1)$ if for every $k$-subset $A$ of its vertices  and every $B\subseteq A$, there exists $x\notin A$ such that $x$ dominates every element of $B$ and every element of $A\setminus B$ dominates $x$. A tournament has property $S_{k}$ if $B=\varnothing $ in the definition before. We give a characterization of those circulant tournaments of prime order having property $P_{2}$ using some results of additive number theory. Some new theoretical results are proved. It is proved that in vertex-transitive doubly regular tournaments properties $S_{3}$ and $P_{3}$ are equivalent and consequently, the Paley tournament $QR_{p}$ has property $P_{3}$ for every $p\equiv 3 \bmod 4$ such that $p\geq 19$. It is also shown that the out- and in-neighborhood of every vertex of $QR_{p}$ induce a circulant tournament with a special structure. As corollaries, we obtain that the out- and in-neighborhood of every vertex of $QR_{p}$ has property $S_{3}$ if and only if $QR_{p}$ has property $S_{4}$ and that $QR_{67}$ has property $S_{4}$. In addition, non-vertex-transitive doubly regular tournaments of Szekeres type are considered. We show that the infinite families of Szekeres tournaments and their converses satisfy property $P_{3}$.