Description
Consider the following question for a fixed triangle T: If the points of a circle are two-colored, can we always find a similar copy of T such that all three vertices have the same color? Surprisingly, the answer is yes for exactly one triangle T. (You have until my talk to guess what this triangle is!)
More generally, Stromquist considered the following question for a k-tuple (d1,..,dk) of positive numbers such that d1+...+dk=1: If the points of a unit perimeter circle are two-colored, can we always find k points of the same color such that the distances of cyclically consecutive points, measured along the arcs, are d1,d2,…,dk in any order? He conjectured that the answer is yes for a specific k-tuple. Our main result is that the answer is no for every other k-tuple.
Our proof goes by establishing a connection to problems in number theory, Beatty sequences and Fraenkel's conjecture. I will also state a neat new conjecture about balancing a certain sequence of numbers which would imply Stromquists's conjecture.
Joint work with Gábor Damásdi, Nóra Frankl and János Pach.
https://arxiv.org/abs/2504.10687