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A central theorem in combinatorics is Sperner's Theorem, which determines themaximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does notcontain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this resultand determined the largest family not containing a $k$-chain $F_1\subsetneq\ldots \subsetneq F_k$. Erd\H{o}s and Katona and later Kleitman asked how manysuch chains must appear in families whose size is larger than the correspondingextremal result.  This question was resolved for $2$-chains by Kleitman in $1966$, who showedthat amongst families of size $M$ in $\mathcal{P}(n)$, the number of $2$-chainsis minimized by a family whose sets are taken as close to the middle layer aspossible. He also conjectured that the same conclusion should hold for all $k$,not just $2$. The best result on this question is due to Das, Gan and Sudakovwho showed that Kleitman's conjecture holds for families whose size is at mostthe size of the $k+1$ middle layers of $\mathcal{P}(n)$, provided $k\leq n-6$.Our main result is that for every fixed $k$ and $\epsilon>0$, if $n$ issufficiently large then Kleitman's conjecture holds for families of size atmost $(1-\epsilon)2^n$, thereby establishing Kleitman's conjectureasymptotically. Our proof is based on ideas of Kleitman and Das, Gan andSudakov. Several open problems are also given.

Kleitman's conjecture about families of given size minimizing the number of $k$-chains