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Statement

Context: the paper's definition of intersector sets and Sárközy's theorems, quoted there, that the squares and the shifted primes $p-1$ are difference intersector sets.

The examples (p. 209, unlabeled). The paper states that neither sequence is a sum intersector set.

• For $A=\{1,4,7,\ldots,3k+1,\ldots\}$ one has $A(N)/N\ge1/3$, but $a_x+a_y=z^2$ (16) is not solvable.
• For $A=\{4,7,\ldots,3k+1,\ldots\}$ one has $A(N)/N\ge1/3-1/N$, but $a_x+a_y=p-1$ (17) is not solvable.

The paper gives no reason. Every sum $a_x+a_y$ is $\equiv2\pmod3$, while a square is $\equiv0$ or $1\pmod3$; and $p-1\equiv2\pmod3$ forces $p=3$, so $p-1=2$, which is not a sum of two elements at least $4$ (an observation of this page). Both conclusions allow $x=y$.

The guess (p. 209, quoted). "We guess that these examples are extremal in the sense that for $\varepsilon>0$, $N>N_0(\varepsilon)$, $\frac{A(N)}{N}>\frac13+\varepsilon$ implies the solvability of both equations (16) and (17)."

The guess concerns sets $A$ of positive integers up to $N$, as in the paper's finite setting. The paper offers it as a guess and proves nothing toward it.

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: p. 209, the paragraph after the proof of Theorem 3. The edition read is identified on the source card.

Read depth. Claims checked: the examples and the guess were read clause by clause on the printed page. Nothing here is independently reviewed.

Bears on

• Problem 438: the problem asks how large $A\subseteq\{1,\ldots,N\}$ can be when $A+A$ contains no square. The first example gives such sets with at least $N/3$ elements, and the guess, for equation (16), would make the answer $(1/3+o(1))N$. The problem page records the answer $(11/32+o(1))N$, and its claim page states that Massias's construction of density $11/32$ shows the guess false for (16). The guess for (17) is not part of the problem.
• Problem 439: the problem asks for a monochromatic pair $x\ne y$ with $x+y$ a square in every finite coloring of the integers. The first example is a set of density $1/3$ with no square sum, the density-side fact recorded beside the problem's source key; the paper does not pose the coloring question.