There are many facts
known about the size of subsets of certain kinds in free lattices and free
products of lattices. Examples: every chain in a free lattice is at most countable;
every “large” subset contains an independent set; if the free product of a
set of lattices contains a “long” chain, so does the free product of a flnite
subset of this set of lattices. Here we investigate these problems in the setting
of a variety V of m-lattices, where m is an infinite regular cardinal. An
m-lattice L is a lattice in which for any nonempty set S with |S| < m, the
meet and join exist in L. We obtain generalizations of many finitary results
to the m-complete case. Our basic set-theoretic tool is the Erdös-Rado
theorem.
