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Statement

Notation (p. 204): $\Gamma(N)$ is the set of the subsets of $\{1,\ldots,N\}$ and $A(N)$ counts the elements of $A$ up to $N$.

Theorem 5 (p. 212, quoted with its displays). "If $k$, $d$ are positive integers, $\varepsilon>0$ is any real number, $N>N_0(k,d,\varepsilon)$ and we put $B=\{b_1,b_2,\ldots,b_k\}=\{d,2d,\ldots,kd\}$, then $A\subset\Gamma(N)$ and"

$$A(N)>\Bigl(\frac1{k+1}+\varepsilon\Bigr)N\qquad(28)$$

"imply the solvability of"

$$a_x-a_y=b_z.\qquad(29)$$

Role in the paper (p. 212). Section 4 opens by stating that, as $N\to+\infty$, there are difference intersector sets $B\subset\{1,\ldots,N\}$ with $B(N)$ bounded, while for sum intersector sets $B(N)\to+\infty$ must hold, and calls the first statement "near trivial". Theorem 5 is the first statement: for fixed $k$ and $d$, the $k$-element set $B$ satisfies the finite form of the definition for every density $\varepsilon>1/(k+1)$ in (3), by Theorem 5 applied with $\varepsilon-1/(k+1)$ in place of $\varepsilon$ (an observation of this page on how the two statements match). Theorem 6 is the second. Section 6 (p. 222) uses Theorem 5 to show that a union of blocks of consecutive integers is a difference intersector set.

Source. P. Erdős and A. Sárközy, On differences and sums of integers, II, Bull. Soc. Math. Grèce (N.S.) 18 (1977), no. 2, 204--223: the statement on p. 212, the proof on pp. 212--213. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 212--213. Cover $\{1,\ldots,N\}$ by blocks of $k+1$ consecutive terms of an arithmetic progression of difference $d$, one family of blocks for each residue $r$ modulo $d$, as in (30). By (28), for large $N$ some block holds two elements of $A$; their difference is $jd$ with $1\le j\le k$, an element of $B$.