Improved bound in the proof for k=5 and arbitrary l

In their proof of the case $ k=5 $ and arbitrary $ \ell $, Abel et al. [ABBCDHKLPW] proved a doubly exponential upper bound on the number $ p(\ell) $ of points that guarantees the occurrence of an $ \ell $-tuple of collinear points or a $ 5 $-tuple of points with no other point in their convex hull (an empty pentagon). The upper bound on $ p(\ell) $ was improved by Barát et al. [BDJPSSVW] to $ p(\ell) \le 328\ell^2 $.

[BDJPSSVW] János Barát, Vida Dujmović, Gwenaël Joret, Michael S. Payne, Ludmila Scharf, Daria Schymura, Pavel Valtr, and David R. Wood. Empty pentagons in point sets with collinearities, arxiv:1207.3633, 2012.


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