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.

## 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.