  A classical result of Bondy and Simonovits in extremal graph theory states
that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at
most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for
$k=2,3,5$.
  In this paper we study ordered variants of this problem and prove some tight
estimates for a certain class of ordered cycles that we call bordered cycles.
In particular, we show that the maximum number of edges in an ordered graph
avoiding bordered cycles of length at most $2k$ is $\Theta(n^{1+1/k})$.
  Strengthening the result of Bondy and Simonovits in the case of 6-cycles, we
also show that it is enough to forbid these bordered orderings of the 6-cycle
to guarantee an upper bound of $O(n^{4/3})$ on the number of edges.
