Let c denote the cardinality of the continuum. For ordinals α, β and a cardinal c, the 3-uniform ordinal Ramsey property α→(β,c)23 is defined as follows: for every 2-coloring of the 3-element subsets of α (identified with its ordinal type), either there exists a subset of order type β all of whose 3-element subsets are colored red, or there exists a subset of cardinality c all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal β and every integer n≥2, the partition relation c→(β,n)23 holds.