Abstract
It is proved that for every finite latticeL there exists a finite latticeL′ such that for every partition of the points ofL′ into two classes there exists a lattice embeddingf:L→L′ such that the points off(L) are in one of the classes.
This property is called point-Ramsey property of the class of all finite lattices. In fact a stronger theorem is proved which implies the following: for everyn there exists a finite latticeL such that the Hasse-diagram (=covering relation) has chromatic number >n. We discuss the validity of Ramseytype theorems in the classes of finite posets (where a full discussion is given) and finite distributive lattices. Finally we prove theorems which deal with partitions of lattices into an unbounded number of classes.
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Nešetřil, J., Rödl, V. Combinatorial partitions of finite posets and lattices —Ramsey lattices. Algebra Universalis 19, 106–119 (1984). https://doi.org/10.1007/BF01191498
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DOI: https://doi.org/10.1007/BF01191498