Note on Existence and Non-Existence of Large Subsets of Binary Vectors with Similar Distances

We consider vectors from {0, 1}. The weight of such a vector v is the sum of the coordinates of v. The distance ratio of a set L of vectors is dr(L) := max{ρ(x, y) : x, y ∈ L}/min{ρ(x, y) : x, y ∈ L, x 6= y}, where ρ(x, y) is the Hamming distance between x and y. We prove that (a) there are no positive constants α and C such that every set K of vectors with weight p contains a subset K′ with |K′| ≥ |K|α and dr(K′) ≤ C, even when |K| ≥ 2, (b) for a set K of vectors with weight p, and a constant C > 2, there exists K′ ⊆ K such that dr(K) ≤ C and K′ ≥ |K|α, where α = 1/⌈log(p/2)/ log(C/2)⌉.