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IV
Sequences without Arithmetic

Progressions

The gist of the result of Chapter IV is that a sequence of integers
with “positive density” must contain an arithmetic progression (of at
least three distinct terms).
More precisely and in sharper, finitized form, this is the statement
that, if 1> 0, then for large enough n, any subset of the nonnegative
integers below n with at least 1n members must contain three terms
a, b, c where a< b < c and a + c 2b. This is a shock to nobody.
If a set is “fat” enough, it should contain all sorts of patterns. The
shock is that this is so hard to prove.
At any rate we begin with a vastly more general consideration, the
notion of an “affine property” of finite sets of integers. So let us agree
to call a property P an affine property if it satisfies the following two
conditions:

1. For each fixed pair of integers α, β with α ø 0, the set A(n) has
P if and only if αA(n) + β has P.
2. Any subset of a set, which has P, also has P.

Thus, for example, the property P A of not containing any arith-
metic progressions is an affine property. Again the trivial property
P 0 of just being any set is an affine one.
Now we fix an affine property P and consider a largest subset of
the nonnegative integers below n, which has P. (Thus we require
that this set has the most members possible, not just to be maximal.)
There may be several such sets but we choose one of them and denote
it by S(n;Pð . We also denote the number of elements of this set by

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