On subspaces spanned by random selections of plus/minus 1 vectors

The work of Kanter and Sompolinsky [S] on associative memories gives rise to the following question. Let the vectors vl, . . . . v, be chosen randomly from { f 11” (the + 1 vectors of length n). What is the probability that the subspace spanned by v, , . . . . v, over the reals contains a f 1 vector different from fv,, . . . . f v,? (The reals can be replaced by any field of characteristic zero, since the answers are the same.) Some of the results of [S] seemed at first to suggest that if p, n --, cc while p/n + u for some a, 0 < tl < 1, then this probability might tend to 0 for CI < 1 2/7r and might tend to 1 for c( > 1-2/n. However, G. Kalai and N. Linial (unpublished) conjectured that this is not the case, and that in fact this probability is dominated by the probability that some 3 of the vi have a linear combination that is a + 1 vector different from the *vi. This paper proves this conjecture. 124 0097-3165/88 33.00