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We address a supersaturation problem in the context of forbidden subposets. Afamily $\mathcal{F}$ of sets is said to contain the poset $P$ if there is aninjection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p)\subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \le c,d$ iscalled butterfly. The maximum size of a family $\mathcal{F} \subseteq 2^{[n]}$that does not contain a butterfly is $\Sigma(n,2)=\binom{n}{\lfloor n/2\rfloor}+\binom{n}{\lfloor n/2 \rfloor+1}$ as proved by De Bonis, Katona, andSwanepoel. We prove that if $\mathcal{F} \subseteq 2^{[n]}$ contains$\Sigma(n,2)+E$ sets, then it has to contain at least $(1-o(1))E(\lceil n/2\rceil +1)\binom{\lceil n/2\rceil}{2}$ copies of the butterfly provided $E\le2^{n^{1-\varepsilon}}$ for some positive $\varepsilon$. We show by aconstruction that this is asymptotically tight and for small values of $E$ weshow that the minimum number of butterflies contained in $\mathcal{F}$ isexactly $E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$.

Supersaturation and stability for forbidden subposet problems