How many random edges make a dense hypergraph non‐2‐colorable?

Abstract

AbstractWe study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. (Random Struct Algorithms 22 (2003) 33–42), and continued in (Bohman et al., Random Struct Algorithms 24 (2004) 105–117) and (Krivelevich et al., Random Struct Algorithms 29 (2006), 180–193). Here we obtain a tight bound on the number of random edges required to ensure non‐2‐colorability. We prove that for any k‐uniform hypergraph with Ω(nk‐ε) edges, adding ω(nkε/2) random edges makes the hypergraph almost surely non‐2‐colorable. This is essentially tight, since there is a 2‐colorable hypergraph with Ω(nk‐ε) edges which almost surely remains 2‐colorable even after adding o(nkε/2) random edges.© 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008