  A famous theorem of Erdos and Szekeres states that any sequence of $n$
distinct real numbers contains a monotone subsequence of length at least
$\sqrt{n}$. Here, we prove a positive fraction version of this theorem. For $n
> (k-1)^2$, any sequence $A$ of $n$ distinct real numbers contains a collection
of subsets $A_1,\ldots, A_k \subset A$, appearing sequentially, all of size
$s=\Omega(n/k^2)$, such that every subsequence $(a_1,\ldots, a_k)$, with $a_i
\in A_i$, is increasing, or every such subsequence is decreasing. The
subsequence $S = (A_1,\ldots, A_k)$ described above is called block-monotone of
depth $k$ and block-size $s$. Our theorem is asymptotically best possible and
follows from a more general Ramsey-type result for monotone paths, which we
find of independent interest. We also show that for any positive integer $k$,
any finite sequence of distinct real numbers can be partitioned into $O(k^2\log
k)$ block-monotone subsequences of depth at least $k$, upon deleting at most
$(k-1)^2$ entries. We apply our results to mutually avoiding planar point sets
and biarc diagrams in graph drawing.
