From Don Knuth‘s Theory Lunch Talk at Stanford on Oct 6. I found a blog post talking about the exact same thing, so won’t repeat. The idea behind the proof of this rather combinatorial theorem is interesting. The main idea is to construct a nice mapping between a permutation of (1,2, …., n) and a pair of (finite) sets: (A, B) based on the elements inside the sets.

You can try to figure out and prove when the equality holds by yourself as an exercise.

When is this Theorem useful? For example, in making representative families <– so that we have better memory performance. Think about it!

Too lazy D:< Plzzzz let me know if you made a nice drawing for the proof of the theorem (Shouldn’t be too hard) so that I can include your image here rather than this textbook cover.

The following theorem of Bollobas is an important result from extremal set theory. It has apparently been rediscovered several times by others. It implies certain classical results, such as Sperner’s Theorem, and has some interesting extensions.

**Theorem** (Bollobas). Let $latex A_1, ldots, A_m$ and $latex B_1, ldots, B_m$ be sequences of $latex a$-element and $latex b$-element sets respectively. Suppose that $latex A_i cap B_i = emptyset$ and $latex A_i cap B_j neq emptyset$ whenever $latex i neq j$. Then $latex m leq {a+b choose a}$.

*Proof*. Let $latex X$ be the union of all the $latex A_i$’s and $latex B_i$’s. Consider an arbitrary permutation $latex pi$ of $latex X$. We claim that there is at most one pair $latex (A_i, B_i)$ such that all of the elements of $latex A_i$ precede those of $latex B_i$ in the ordering $latex pi$. Suppose to the contrary that there are two…

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