In the book we gave two proofs of this theorem (Thm. 8.8):
Bollobás's Two Families Theorem: Let $A_1,\ldots,A_m$ and $B_1$,$\ldots$, $B_m$ be two sequences of sets such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Then \[ \sum_{i=1}^m {{a_i+b_i}\choose{a_i}}^{-1}\leq 1, \] where $a_i=|A_i|$ and $b_i=|B_i|$.This theorem was originally stated in a different way:
Lemma in [1]: Let $A_1,\ldots,A_m$ and $B_1$,$\ldots$, $B_m$ be two sequences of subsets of $[n]$ such that $A_i\cap B_i=\emptyset$ for all $i$, and $A_j\not\subseteq A_i\cup B_i$ for all $i\neq j$. Then \[ \sum_{i=1}^m {{n-b_i}\choose{a_i}}^{-1}\leq 1. \]It is not immediately clear how to derive the theorem from this lemma.
Therefore, the theorem was offered (later) as a conjecture by A. Ehrenfeucht and J. Mycielski, and it was proved by G.O.H. Katona [2] who gave a beautiful proof (in the uniform case) using Lubell's method of counting permutations -- this is the second proof of Thm. 8.8 in the book. Independently, this form was also proved by F. Jaeger and C. Payan [3]. The general (non-uniform) case of Thm. 8.8 was published by T.G. Tarján [4], a student of Katona then.
So, Thm. 8.8 in the book should be called the "Bollobás-Katona two families theorem".
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