We generalize Kada’s definable strengthening of Dilworth’s characterization of the class of quasi-orders admitting an antichain of a given finite cardinality.

We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.