Non-uniform Turلn-type problems

Given positive integers n , k , t , with 2 ⩽ k ⩽ n , and t < 2 k , let m ( n , k , t ) be the minimum size of a family F of (nonempty distinct) subsets of [ n ] such that every k-subset of [ n ] contains at least t members of F , and every ( k - 1 ) -subset of [ n ] contains at most t - 1 members of F . For fixed k and t, we determine the order of magnitude of m ( n , k , t ) . We also consider related Turán numbers T ⩾ r ( n , k , t ) and T r ( n , k , t ) , where T ⩾ r ( n , k , t ) ( T r ( n , k , t ) ) denotes the minimum size of a family F ⊂ [ n ] ⩾ r F ⊂ [ n ] r such that every k-subset of [ n ] contains at least t members of F . We prove that T ⩾ r ( n , k , t ) = ( 1 + o ( 1 ) ) T r ( n , k , t ) for fixed r , k , t with t ⩽ k r and n → ∞ .