Kalai–Linial–Odlyzko sharp failure-probability conjecture

Determine whether, for independent uniformly random vectors v_1,\ldots,v_r\in\{-1,1\}^n, the largest r for which \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n=\{\pm v_1,\ldots,\pm v_r\} with probability 1-o(1) is r=n-1 and whether the corresponding failure probability is asymptotic to 4\binom{r}{3}(3/4)^n.

Background

For independent random sign vectors, the paper studies when their linear span contains no Boolean-cube vertices beyond the sampled vectors and their negatives. Earlier work established that the desired event holds with high probability for ranges of r substantially below n, while Komlós’s result shows that the probability tends to zero when r=n.

The Kalai–Linial–Odlyzko conjecture predicts both the sharp threshold r=n-1 and the precise asymptotic failure probability. The paper proves the threshold with a weaker polynomial error term, thereby confirming only a weaker form of the conjecture and leaving the stated sharp asymptotic unresolved.