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Boolean Lattices: Ramsey Properties and Embeddings

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Abstract

A subposet Q ′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q ′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P ′, let the poset Ramsey number R(P, P ′) be the smallest N such that no matter how the elements of the Boolean lattice Q N are colored red and blue, there is a copy of P with all red elements or a copy of P ′ with all blue elements. We provide some general bounds on R(P, P ′) and focus on the situation when P and P ′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Q n in Q N and for a multicolor version of a poset Ramsey number.

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  1. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Maria Axenovich & Stefan Walzer

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  1. Maria Axenovich
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  2. Stefan Walzer
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Correspondence to Maria Axenovich.

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Axenovich, M., Walzer, S. Boolean Lattices: Ramsey Properties and Embeddings. Order 34, 287–298 (2017). https://doi.org/10.1007/s11083-016-9399-7

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