Exercise Solutions

Linear Algebra Methods in Combinatorics This file contains solutions to some of the exercises, and it will be periodically updated. 1 Exercise Set 2 Exercise 4 (One-Distance Sets). A regular simplex is a set of n + 1 points in R n such that any two points are at distance 1. Prove that no set with this property can have more points. n are such that d(s i , s j) = 1 for all i = j (1) where d() is the Euclidean distance, then m ≤ n + 1. If you cannot prove this bound, try to prove the simpler bound m ≤ n + 2. Warm up (" n + 2 "). We can assume wlog that d() is the square of the Euclidean distance (this will not change the problem), that is d(x, y) = k