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Source. Proposition 6, p. 7, of Ben Green, The Cameron-Erdős conjecture, Bull. London Math. Soc. 36 (2004), no. 6, 769--778, cited from the arXiv manuscript math/0304058v1 (4 April 2003) whose pages the labels below follow, as identified on the source card.

Statement

An additive triple in a set is a triple $(x,y,z)$ of its elements with $x+y=z$ (p. 2). Fix a prime $p\in[2N,4N]$ and regard $[N]=\{1,\ldots,N\}$ as a subset of $\mathbb Z/p\mathbb Z$. The paper sets

$$\epsilon=(\log N)^{-1/11},\qquad M=\bigl\lfloor N\exp\bigl(-(\log N)^{1/12}\bigr)\bigr\rfloor$$

(p. 7). For each $d\in(\mathbb Z/p\mathbb Z)^*$, display (2) (p. 3) partitions $\mathbb Z/p\mathbb Z$ into $M$ arithmetic progressions $I_i$, $i\in\mathbb Z/M\mathbb Z$, of common difference $d$, each of length $L$ or $L-1$ with $L=\lceil p/M\rceil$. The family is built in three steps (p. 7):

• $\mathcal G$ is the collection of all sets that are unions of progressions $I_i$ for some $d$;
• $\mathcal H$ is the collection of members of $\mathcal G$ with at most $\epsilon p^2$ additive triples;
• $\mathcal F$ is the collection of all subsets of $[N]$ obtained by adding at most $\epsilon p$ elements to some $H\cap[N]$ with $H\in\mathcal H$.

Proposition 6 (p. 7). The family $\mathcal F$ has the following properties: (i) every member of $\mathcal F$ has at most $o(N^2)$ additive triples; (ii) every sum-free $A\subseteq[N]$ is contained in some member of $\mathcal F$; (iii) $|\mathcal F|\le2^{o(N)}$.

The $o(\cdot)$ terms are as $N\to\infty$. Part (ii) needs a good length for $A$ (p. 3) to exist with the parameters fixed above; the paper states that one exists "at least for $N$ sufficiently large" and leaves that check to the reader as "easy but slightly tedious" (p. 7, quoted).

Proof pointer

Proof on p. 7. Part (i) follows from the definition of $\mathcal H$ and $p\le4N$; part (iii) counts $p-1$ choices of $d$, $2^M$ unions of progressions and the subsets of $[N]$ of size at most $\epsilon N$. Part (ii) takes the granularization $A'$ of a sum-free $A$ along a good length: Proposition 4 (p. 5), with parameters $\epsilon_1=\epsilon$, $\epsilon_2=\epsilon^2/144$ and $\epsilon_3=\epsilon^2/80$, shows that $A'$ has at most $\epsilon p^2$ additive triples, so $A'\in\mathcal H$, and display (3) (p. 3), $|A'\setminus A|\le\epsilon_1p$, places $A$ in a member of $\mathcal F$. Proposition 4 rests on Proposition 3 (p. 4), and Proposition 5 (p. 6) gives a sufficient condition for a good length to exist. The paper says (p. 2) that this construction was basically achieved in the earlier work of Green and Ruzsa on sum-free sets in $\mathbb Z/p\mathbb Z$ and repeats part of it.

Dependencies

Propositions 3, 4 and 5 (pp. 4--6) and the existence of a good length for the chosen parameters, which the paper asserts without a written check. Read depth: claims checked; the statement and the construction were read on pp. 3 and 7, the proofs for their structure only.

Bears on

• Problem 748: with Proposition 7 (p. 8) it gives Proposition 12 (p. 10), the bound $2^{N/2+o(N)}$ for the number of sum-free subsets of $\{1,\ldots,N\}$, which with the trivial lower bound $2^{\lceil N/2\rceil}$ is the problem's exponent form, and it is the first step toward Theorem 2; on its own it gives no count.