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Ramsey Numbers for Partially-Ordered Sets

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Abstract

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.

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Acknowledgements

The authors would like to thank Mikhail Lavrov for recommending SAT solvers as a method for computing small Boolean Ramsey numbers. Thanks also to Maria Axenovich for discussing previous work on induced Boolean Ramsey numbers.

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, USA

    Christopher Cox

  2. Microsoft, Raleigh, NC, USA

    Derrick Stolee

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  1. Christopher Cox
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  2. Derrick Stolee
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Correspondence to Christopher Cox.

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The authors completed this work while at Iowa State University.

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Cox, C., Stolee, D. Ramsey Numbers for Partially-Ordered Sets. Order 35, 557–579 (2018). https://doi.org/10.1007/s11083-017-9449-9

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