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Extremal Set Theory
Let n be a positive integer and let [n]=1,2,\dots,n. In this section, we consider problems having the following general form: What is the maximum size of a family of subsets of [n] when the family is required to satisfy…
Extremal Set Theory
Let \(n\) be a positive integer and let \([n]=1,2,\dots,n\). In this section, we consider problems having the following general form: What is the maximum size of a family of subsets of \([n]\) when the family is required to satisfy certain properties.
Here is an elementary example.
Example
The maximum size of a family \(\cgF\) of subsets of \([n]\), with \(A\cap B\neq\emptyset\) for all \(A,B\in\cgF\), is \(2^{n-1}\).
For the lower bound, consider the family \(\cgF\) of all subsets of \([n]\) that contain\(1\). Clearly this family has \(2^{n-1}\) elements and any two sets in the family have non-empty intersection.
For the upper bound, let \(\cgF\) be a family of subsets with each pair of sets in \(\cgF\) having non-empty intersection. Then whenever a subset \(S\) is a member of \(\cgF\), the complement \(S'\) of \(S\) cannot belong to \(\cgF\). Since the entire family of all \(2^n\) subsets of \([n]\) can be considered as \(2^{n-1}\) complementary pairs, and at most one set from each pair can belong to \(\cgF\), we conclude that \(|\cgF|\le 2^{n-1}\).
As a second example, we can revisit from and restate the result as follows.
Example
The maximum size of a family \(\cgF\) of subsets of \([n]\) subject to the constraint that when \(A\) and \(B\) are distinct sets in \(\cgF\), then neither is a subset of the other, is \(\binom{n}{\lfloor n/2\rfloor}\).
It is worth noting that in , there is a very small number (one or two) of extremal families, , when \(\cgF\) is a family of subsets of \([n]\), \(|\cgF|= \binom{n}{\lfloor n/2\rfloor}\), and no set in \(\cgF\) is a proper subset of another, then either \(\cgF=\{S\subseteq[n]: |S|=\lfloor n/2\rfloor\}\) or \(\cgF=\{S\subseteq[n]: |S|=\lceil n/2\rceil\}\). And of course, when \(n\) is even, these are exactly the same family.
On the other hand, for , there are many extremal families, since for every complementary pair of sets, either member can be selected.
We close this brief tasting of extremal set theory with a real classic.
Condensed — the full section is in Keller & Trotter, Applied Combinatorics.
Symbols used here
Number of k-element subsets of n things: n!/(k!(n−k)!).
x belongs to A; every element of A is in B.
In either; in both; in A but not B.
Inequalities that allow equality; < and > exclude it.
The two sides are different.
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
Add a_k for k = 1 up to n.
Multiply a_k for k = 1 up to n.
The set with no elements; the number of elements of A.
Questions people ask
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges) — not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
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Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.
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