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In any set of 9 points in $\mathbb{R}^2$ with no triple of points on the same line a convex pentagon exists.

My attempt: I suppose I need to consider some convex hulls. If the convex hull has $\geq 5$ points, just take it. Else consider convex hull of the points inside. Here I don't understand how to finish the "bruteforce".

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  • $\begingroup$ What if all these points are arranged on the same line? Does that count as convex pentagon? $\endgroup$ – Evgeny Mar 17 '17 at 8:08
  • $\begingroup$ Yes, forgot to add that there are no 3 points on the same line. $\endgroup$ – sooobus Mar 17 '17 at 8:13
  • $\begingroup$ Can you prove the milder claim that at a convex quadrilateral exists? $\endgroup$ – Chris Culter Mar 17 '17 at 8:17
  • $\begingroup$ Yes, I can prove that in the set of 5 points a convex quadrilateral exists. $\endgroup$ – sooobus Mar 17 '17 at 8:19
  • $\begingroup$ Okay, you can do sort-of the same thing, but broken into a few different cases depending on ($x$) the number of vertices of the convex hull, ($y$) the number of vertices of the convex hull of the points inside, and ($z$) the number of vertices inside the latter. $\endgroup$ – Chris Culter Mar 17 '17 at 8:45
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Erdos&co solved the happy ending problem by considering the slopes of the sides of the convex envelope and applying the Erdos-Szekeres/Dilworth's theorem: every sequence with $n^2+1$ elements has a monotonic subsequence with $n+1$ elements.

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  • $\begingroup$ I didn't understand: which objects form a sequence? $\endgroup$ – sooobus Mar 17 '17 at 8:45
  • $\begingroup$ According to the Wikipedia article, they only showed that for each $n$-gon there must be a least $m$ such that $m$ points in general position always have an $n$-gon among them. Finding specific least values of $m$ is harder; in particular, the case $n = 5$ was apparently first published in 1970, 35 years later. $\endgroup$ – Mees de Vries Mar 17 '17 at 8:49
  • $\begingroup$ @MeesdeVries: that is true, I should have said "Erdos&co. opened the main road to the solution through the following technique". $\endgroup$ – Jack D'Aurizio Mar 17 '17 at 8:50
  • $\begingroup$ Okay, I know about this problem, but I think that my case should be proven in the easier way. $\endgroup$ – sooobus Mar 17 '17 at 8:54
  • $\begingroup$ @sooobus: I don't think so, but with nowadays computer super-powers the case $m=5$ is easy to crack by brute force, since the space of configurations is not that large. $\endgroup$ – Jack D'Aurizio Mar 17 '17 at 8:56

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