Cheerful news in difficult times: David Conlon and Jeck Lim settled Kupitz’s planar discrepancy problem for pseudolines.

Small updates. I wrote a post about the overwhelming mathematical activities in the last week of April 2023 and the following weeks were as exciting. It was difficult to follow all the activities (not to mention blogging about them). Let me only mention a great lecture by Ingrid Daubechies on “Mathematicians helping art historians and art conservators” and a great four hour seminar lectures by Yuval Wigderson and Eden Kuperwasser on the details of the  3-AP breakthrough by Kelley and Meka. I do plan to blog with some details on the later. This summer I am teaching a summer course in Game Theory. This was a request by students that I found flattering, and I also thought that giving a course will be a nice distraction from current events over here.

Today I want to report about the paper

Everywhere unbalanced configurations, by David Conlon and Jeck Lim

Some years ago I blogged about the following question of Yaakov Kupitz

What is the smallest number C such that for every configuration of n points in the plane there is a line containing two or more points from the configuration for which the difference between the number of points on the two sides of the line is at most C?

Kupitz asked, in particular, if every point set has a line through at least two of its points which is balanced in the sense that the number of points on either side differ by at most some fixed constant c. David Conlon and Jeck Lim showed that if you allow pseudolines rather than lines, the answer is no!

Kupitz’s question extends to pseudolines where it can be formulated using the method of  allowable sequence of permutations introduced by Goodman in Pollack. The best known upper bound was proved by Rom Pinchasi and Rom’s proof uses the method of allowable sequence of permutations and applies to pseudolines. Conlon and Lim’s proof shows that for pseudolines, .

Congratulations David and Jeck!