Supersaturation and stability for forbidden subposet problems

We address a supersaturation problem in the context of forbidden subposets. A family F of sets is said to contain the poset P if there is an injection i : P → F such that p ≤P q implies i(p) ⊂ i(q). The poset on four elements a, b, c, d with a, b ≤ c, d is called a butterfly. The maximum size of a family F ⊆ 2 that does not contain a butterfly is ( n bn/2c ) + ( n bn/2c+1 ) as proved by De Bonis, Katona, and Swanepoel. We prove that if F ⊆ 2 contains ( n bn/2c ) + ( n bn/2c+1 ) +E sets, then it has to contain at least (1−o(1))E(dn/2e+1) (dn/2e 2 ) copies of the butterfly provided E ≤ 2. We show that this is asymptotically tight and for small values of E we show that the minimum number of butterflies contained in F is exactly E(dn/2e+ 1) (dn/2e 2 ) .