SCHWEITZER 1975 Problem 1

Show that there exists a tournament $(T,\rightarrow)$ of cardinality $\aleph_1$ containing no transitive subtournament of size $\aleph_1$ . ( A structure $(T,\rightarrow)$ is a $\textit{tournament}$ if $\rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $(T,\rightarrow)$ is transitive if $\rightarrow$ is transitive, that is, if it orders $T$ .)

A. Hajnal