Homological connectivity of random hypergraphs
  We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction. 
Oliver Cooley, Penny Haxell, Mihyun Kang, Philipp Spr\"ussel
math.CO