# A Positive Fraction Erdos - Szekeres Theorem

## Authors

DOI: 10.1007/PL00009350

- Cite this article as:
- Bárány, I. & P. Valtr Discrete Comput Geom (1998) 19: 335. doi:10.1007/PL00009350

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## Abstract.

We prove a fractional version of the Erdős—Szekeres theorem: for any *k* there is a constant *c*_{k}* > 0* such that any sufficiently large finite set *X⊂***R**^{2} contains *k* subsets *Y*_{1}*, ... ,Y*_{k} , each of size *≥ c*_{k}*|X|* , such that every set *{y*_{1}*,...,y*_{k}*}* with *y*_{i}*ε Y*_{i} is in convex position. The main tool is a lemma stating that any finite set *X⊂***R**^{d} contains ``large'' subsets *Y*_{1}*,...,Y*_{k} such that all sets *{y*_{1}*,...,y*_{k}*}* with *y*_{i}*ε Y*_{i} have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem).
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<onlinepub>26 June, 1998
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