Bondy's Theorem

In mathematics, Bondy's theorem is a theorem in combinatorics that appeared in a 1972 article by John Adrian Bondy. The theorem is as follows:

Let X be a set with n elements and let A₁, A₂, ..., A_n be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets A_i ∩ S are all distinct.

In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct.

From the perspective of computational learning theory, this can be rephrased as follows:

Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C.

This implies that every finite concept class C has its teaching dimension bounded by |C| − 1.