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The idea that 2016 was the worst year in history was already floated back in July, but despite a lot of negativity, there were definitely good things that happened in 2016. This was certainly true in terms of scientific discoveries and breakthroughs. One of my favorites was a significant improvement to the upper bound of the Happy Ending problem after 81 years!

The Happy Ending problem was the brainchild of mathematician Esther Klein, who proved that any five points in general position in the plane must contain a convex quadrilateral.

kleinproof

The extension of the problem asks, for each n, for the minimum number f(n) so that any f(n) points in general position in the plane contain a convex n-gon. Paul Erdös and George Szekeres established an upper bound for f(n) via Ramsey Theory in 1935, a link that highlighted the importance and application of Ramsey Theory to extremal combinatorics and instantly made the problem a classic. It even has connections to computational biology, via it’s link to the Erdös-Szekeres monotone subsequence theorem (published in the same 1935 paper), which in turn is related to the Needleman-Wunsch algorithm.

The “happy ending” associated to the problem comes from the story of Esther Klein and George Szekeres, who fell in love while working on the problem and ended up marrying in 1937, a happy ending that lasted for 68 years (they died within an hour of each other in 2005).

In their 1935 paper, Erdös-Szekeres proved an upper bound for f(n) using elementary combinatorics, namely

f(n) \leq {2n -4 \choose n-2}+1 = 4^{n-o(n)}.

The difficulty of improving the upper bound for f(n) can be appreciated by noting that it took 63 years for there to be any improvement to the Erdös-Szekeres result, and when an improvement did arrive in 1998 it was to remove the “+1” (Fan Chung and Ron Graham, Discrete Geometry 1998) reducing the upper bound to f(n) \leq {2n-4 \choose n-2} for n \geq 4.

Considering that the best lower bound for f(n), conjectured to be optimal, is f(n) \geq 2^{n-2}+1 (proved by Erdös and Szekeres in 1960), the +1 improvement left a very long way to go. A few other improvements after the Chung-Graham paper was published reduced the upper bound some more, but at most by constant factors.

Then, earlier this year, Andrew Suk announced an astonishing improvement. Building on a theorem of Attila Pór and Pavel Valtr, he proved that

f(n) = 2^{n+o(n)}.

The result doesn’t exactly match the conjectured lower bound, so one cannot say the happy ending to the Happy Ending Problem arrived in 2016, but it’s so so so close (i.e. it solves the problem asymptotically) that one can only be left with optimism and hope for 2017.

Happy new year!

 

 

 

 

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