  In one of their seminal articles on allowable sequences, Goodman and Pollack
gave combinatorial generalizations for three problems in discrete geometry, one
of which being the Dirac conjecture. According to this conjecture, any set of
$n$ noncollinear points in the plane has a point incident to at least $c n$
connecting lines determined by the set. The notion of allowable sequences of
permutations provides a natural combinatorial setting for analyzing these
problems. Within this formalism, the conjectured generalization reads as
follows: \emph{Any nontrivial allowable $n$-sequence $\Sigma$ has a local
sequence $\Lambda_i$ whose half-period is at least $c n$.} The conjecture is
confirmed here with a concrete bound $c=1/845$. Several related problems are
discussed.
