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{\bf A.Odlyzko,J.B.Shearer, and R.Siders }
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Monotonic subsequences in dimensions higher than one.
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The 1935 result of Erdos and Szekeres that any sequence 
of at least  $n^{2}+1$ real numbers contains a monotonic 
subsequence of at least $n+1$ terms has stimulated
 extensive furher research, including a  paper of
 J.B.Kruskal that defined an extension of monotonicity 
for higher dimensions. This paper provides a proof of
 a weakened form of Kruskal's conjecture for 2 dimensional 
Euclidean space by showing that there exists a sequence
 of n points in the plane for which the longest monotonic
 subsequences have length $n^{2}+2$  or less.. Weaker results
 are also obtained for higher dimensions. The average 
length of the longest increasing monotonic subsequence
is shown  to be $\sim 2n^{1/2}$ as
 $n\rightarrow\infty$ for each dimension.
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