Maximum size t-cross-intersecting and intersecting families with degree conditions Yongbin Ou We present four main results: (1) A solution to the problem of finding two set systems A and B such that A is r 1-intersecting, B is r 2-intersecting, A, B are t-cross-intersecting and |A| + |B| is a maximum; (2) A solution to the problem of finding two set systems A and B such that A, B are Sperner, t-cross-intersecting and |A| + |B| is a maximum; (3) A solution to the problem of finding the maximum size of an intersecting set system F such that the complementary degree c(F) = s for a specified value s; (4) An asymptotic result on the complementary degree of an intersecting set system.