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