In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to in which no member of the set is a multiple of any other must have a logarithmic density that goes to zero as becomes large. The theorem is named after Felix Behrend, who published it in 1935.
Statement
The logarithmic density of a set of integers from 1 to can be defined by setting the weight of each integerto be, and dividing the total weight of the set by . The resulting number is 1 or close to 1 when the set includes all of the integers in that range, but smaller when many integers are missing, and particularly when the missing integers are themselves small. A subset of is called primitive if it has the property that no subset element is a multiple of any other element. Behrend's theorem states that the logarithmic density of any primitive subset must be small. More precisely, the logarithmic density of such a set must be. For infinite primitive sequences, the maximum possible density is smaller,.
Examples
There exist large primitive subsets of. However, these sets still have small logarithmic density.
In the subset, all pairs of numbers are within a factor of less than two of each other, so no two can be multiples. It includes approximately half of the numbers from to. By Dilworth's theorem this subset has maximum cardinality among all subsets in which no two are multiples. But because all of its elements are large, this subset has low logarithmic density, only.
Both of these subsets have significantly smaller logarithmic density than the bound given by Behrend's theorem. Resolving a conjecture of G. H. Hardy, both Paul Erdős and Subbayya Sivasankaranarayana Pillai showed that, for, the set of numbers with exactly prime factors has logarithmic density exactly matching the form of Behrend's theorem. This example is best possible, in the sense that no other primitive subset has logarithmic density with the same form and a larger leading constant.
History
This theorem is known as Behrend's theorem because Felix Behrend proved it in 1934, and published it in 1935. Paul Erdős proved the same result, on a train ride in 1934 in which he traveled from Hungary to Cambridge to escape the growing anti-semitism of Europe at that time, but on his arrival he discovered that Behrend's proof was already known.
