 
Summary: On the 2Colorability of Random Hypergraphs
Dimitris Achlioptas and Cristopher Moore
1
Microsoft Research, Redmond, Washington optas@microsoft.com
2
Computer Science Department, University of New Mexico, Albuquerque and the
Santa Fe Institute, Santa Fe, New Mexico moore@cs.unm.edu
Abstract. A 2coloring of a hypergraph is a mapping from its vertices to
a set of two colors such that no edge is monochromatic. Let Hk(n, m) be
a random kuniform hypergraph on n vertices formed by picking m edges
uniformly, independently and with replacement. It is easy to show that if
r rc = 2k1
ln 2  (ln 2)/2, then with high probability Hk(n, m = rn)
is not 2colorable. We complement this observation by proving that if
r rc  1 then with high probability Hk(n, m = rn) is 2colorable.
1 Introduction
For an integer k 2, a kuniform hypergraph H is an ordered pair H = (V, E),
where V is a finite nonempty set, called the set of vertices of H, and E is a
family of distinct ksubsets of V , called the edges of H. For general hypergraph
terminology and background see [5]. A 2coloring of a hypergraph H = (V, E)
