Partial order on a family of kk-subsets of a linearly ordered set

For kk-subsets A,BA,B of the rationals QQ, define A≻nBA≻nB if a>ba>b holds for at least nn ordered pairs (a,b)∈A×B(a,b)∈A×B, where k,nk,n are integers, 1⩽n⩽k21⩽n⩽k2. We prove that (1) the relation ≻n≻n is transitive if and only if k2-k+1⩽nk2-k+1⩽n, and (2) there is a cyclic sequence A1≻nA2≻n⋯≻nAr≻nA1A1≻nA2≻n⋯≻nAr≻nA1 of kk-subsets of QQ if and only if 1⩽n⩽k2-⌊(k+1)2/4⌋1⩽n⩽k2-⌊(k+1)2/4⌋. We also investigate the length of such cyclic sequences.