The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex $$Y_2(n,p)$$ was first studied by Babson, Hoffman, and Kahle. They proved that threshold probability for simple connectivity is about $$p\approx n^{-1/2}$$ . In this paper, we show at most $$p\le (\gamma n)^{-1/2}$$ , where $$\gamma =4^4/3^3$$ conjecture sharp. fact, $$p=(\gamma a sharp stronger property every ...

Abstract Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p , sampled according to one of the following two models: either every entry is independently taken be probability (the ‘Bernoulli’ model) or each row uniformly from set all length- n vectors exactly pn ones ‘combinatorial’ model). We give simple proofs (essentially best-possible) fact that in both models, if $\min(p,1-p)\ge...