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Keyword: hypergraph coloring ×Clear all

4 claims from 2 papers

1. Computer Science › Constraint Satisfaction and Optimization

   On the Solution-Space Geometry of Random Constraint Satisfaction Problems

   Achlioptas and Ricci‐Tersenghi · arXiv (Cornell University) · 2006

   The paper studies how the set of solutions to random constraint problems changes as constraints are added, to understand why known fast algorithms fail at densities well below where solutions stop existing.

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   1. UncheckedIn random constraint problems, well before solutions vanish, they split into very many small clusters that lie far apart from one another.“In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters.”
   2. UncheckedIn these random constraint problems, within each cluster of solutions most variables are frozen, meaning they take only one value.“Moreover, inside each cluster most variables are frozen, i.e., take only one value.”

2. Mathematics › Limits and Structures in Graph Theory

   Hypergraph Ramsey numbers

   Conlon, Fox and Sudakov · arXiv (Cornell University) · 2008

   The paper gives new upper and lower bounds for several hypergraph Ramsey numbers, including r_3(s,n) and the three-colour number r_3(n,n,n), and makes progress on related Ramsey-type problems.

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   1. UncheckedFor colourings of triples, the paper gives a new upper bound on r_3(s,n) of 2^(n^(s-2) log n), with s fixed, improving the exponent of Erdős and Rado's 1952 bound.“In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952.”
   2. Unchecked“We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n.”