### On the number of discrete chains in the plane

Combinatorics Seminar

3rd December 2019, 11:00 am – 12:00 pm

Fry Building, G.07

Determining the maximum number of unit distances that can be spanned by n points in the plane is a difficult problem, which is wide open. The following more general question was recently considered by Eyvindur Ari Palsson, Steven Senger, and Adam Sheffer. For given distances t_1,...,t_k a (k+1)-tuple (p_1,...,p_{k+1}) is called a k-chain if ||x_i-x_{i+1}||=t_i for i=1,...,k. What is the maximum possible number of k-chains that can be spanned by a set of n points in the plane? Improving the result of Palsson, Senger and Sheffer, we determine this maximum up to a small error term (which, for k=1 mod 3 involves the maximum number of unit distances). We also consider some generalisations, and the analogous question in R^3. Joint work with Andrey Kupvaskii.

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