 
Summary: The number of orientations having no fixed tournament
Noga Alon
Raphael Yuster
Abstract
Let T be a fixed tournament on k vertices. Let D(n, T) denote the maximum number of
orientations of an nvertex graph that have no copy of T. We prove that D(n, T) = 2tk1(n)
for all sufficiently (very) large n, where tk1(n) is the maximum possible number of edges of a
graph on n vertices with no Kk, (determined by TurŽan's Theorem). The proof is based on a
directed version of SzemerŽedi's regularity lemma together with some additional ideas and tools
from Extremal Graph Theory, and provides an example of a precise result proved by applying
this lemma. For the two possible tournaments with three vertices we obtain separate proofs that
avoid the use of the regularity lemma and therefore show that in these cases D(n, T) = 2 n2
/4
already holds for (relatively) small values of n.
1 Introduction
All graphs considered here are finite and simple. For standard terminology on undirected and
directed graphs the reader is referred to [4]. Let T be some fixed tournament. An orientation of an
undirected graph G = (V, E) is called Tfree if it does not contain T as a subgraph. Let D(G, T)
denote the number of orientations of G that are Tfree. Let D(n, T) denote the maximum possible
value of D(G, T) where G is an nvertex graph. In this paper we determine D(n, T) precisely
