On the size of coset unions

Let g1H1, … , gnHn be cosets of subgroups H1, … , Hn of a finite group G such that g1H1∪ … ∪ gnHn≠ G. We prove that | g1H1∪ … ∪ gnHn| ≤ γn| G| where γn< 1 is a constant depending only on n. In special cases, we show that γn= (2 n- 1 ) / 2 n is the best possible constant with this property and we conjecture that this is generally true