Random hypergraphs and property B

In 1964 Erd\H{o}s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it known whether there exist such $k$-graphs with smaller number of edges. Erd\H{o}s' bound consequence the fact hypergraph $k^2/2$ vertices and $M(k)=(1+\oh{1}) randomly chosen size $k$ asymptotically almost surely Our first main result implies for any $\varepsilon > 0$, $(1-\varepsilon) M(k)$ uniformly a.a.s. The presented proof an adaptation second moment method analogous developments Achlioptas Moore from 2002 who considered problem fixed tending infinity. part paper we consider algorithmic coloring random $k$-graphs. We show quite simple, somewhat greedy procedure, finds proper on vertices, at most $\Oh{k\ln k\cdot 2^k}$ That same asymptotic order as analogue \emph{algorithmic barrier} defined by Coja-Oghlan in 2008, case $k$.