99 research outputs found
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
The condensation transition in random hypergraph 2-coloring
For many random constraint satisfaction problems such as random
satisfiability or random graph or hypergraph coloring, the best current
estimates of the threshold for the existence of solutions are based on the
first and the second moment method. However, in most cases these techniques do
not yield matching upper and lower bounds. Sophisticated but non-rigorous
arguments from statistical mechanics have ascribed this discrepancy to the
existence of a phase transition called condensation that occurs shortly before
the actual threshold for the existence of solutions and that affects the
combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi,
Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time
that a condensation transition exists in a natural random CSP, namely in random
hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment
method breaks down strictly \emph{before} the condensation transition. Our
proof also yields slightly improved bounds on the threshold for random
hypergraph 2-colorability. We expect that our techniques can be extended to
other, related problems such as random k-SAT or random graph k-coloring
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