Multiply-intersecting families revisited

Motivated by the Franklʹs results in [P. Frankl, Multiply-intersecting families, J. Combin. Theory B 53 (1991) 195–234], we consider some problems concerning the maximum size of multiply-intersecting families with additional conditions. Among other results, we show the following version of the Erdős–Ko–Rado theorem: for all r ⩾ 5 and 1 ⩽ t ⩽ 2 r + 1 − 3 r − 1 there exist positive constants ε and n 0 such that if n > n 0 and | k n − 1 2 | < ε then r-wise t-intersecting k-uniform families on n vertices have size at most max { ( n − t k − t ) , ( t + r ) ( n − t − r k − t − r + 1 ) + ( n − t − r k − t − r ) } .