A problem of Kusner on equilateral sets

R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983), 196–199] asked whether a set of vectors in a d-dimensional real vector space such that the l-p distance between any pair is 1, has cardinality at most d+1. We show that this is true for p=4 and any d >= 1, and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most $(2\lceil p/4\rceil-1)d+1$ifpisaneveninteger,andatleast$(1+\epsilon_p)d$ if 1<p<2, where ${ϵ}_p>0$ depends on p.