---

Statement

Lemma 2 (p. 95). Let $n\ge N_0$ and let $R(n)=\bigcup_{i=1}^{r(n)}R_i(n)$ be a set of integers such that

• (2a) $\lvert R_i(n)\rvert\in\{1,2\}$;
• (2b) $\lvert R_i(n)\rvert=1$ for at most one $i$;
• (2c) $R_i(n)\cap R_j(n)=\emptyset$ for $1\le i<j\le r(n)$;
• (2d) $R_i(n)\ne R_k(m)$ for all $1\le i\le r(n)$, $1\le k\le r(m)$, $m\ne n$;
• (2e) $r(n)>c\log n$ for some constant $c>\log^{-1}(4/3)$ and all $n\ge N_1$.

Then there is a number $N_2$ such that for all $n\ge N_0$ there is a set $X(n)\subseteq R(n)$ with (2f) $\lvert X(n)\rvert=r(n)$, (2g) $\lvert X(n)\cap R_i(n)\rvert=1$ for every $i\le r(n)$, and (2h) for every $m\ge N_2$, $m\ne n$, some $j\le r(m)$ has $X(n)\cap R_j(m)=\emptyset$.

Remark (pp. 96--97). The paper's typical application: for a sequence $A$, let $R_i(n)=\{a_j,a_k\}$ run over the representations $n=a_j+a_k$, $a_j\le a_k$. Then (2a) to (2d) hold automatically, and (2e) holds when every large $n$ has at least $c\log n$ representations with $c>\log^{-1}(4/3)$. Deleting $X(n)$ from $A$ destroys every representation of $n$, while every $m\ge N_2$, $m\ne n$, stays in $2(A\setminus X(n))$.

Proof pointer

Pp. 95--96. Choose $\delta>0$ with $c\log(4/3)=1+\delta$ and $N_2\ge N_1$ with $\sum_{m\ge N_2}m^{-1-\delta}<1/2$. By Lemma 1, for each $m$ at most $2^{r(n)}(3/4)^{r(m)}$ transversals of $R(n)$ meet every set of $R(m)$; summing over $m\ge N_2$ leaves fewer than $2^{r(n)-1}$ bad transversals, while (2a) and (2b) give at least $2^{r(n)-1}$ transversals in all.

Read depth

Claims checked: the statement and the remark were read clause by clause on the page images of the print, and the proof was followed. Nothing here is independently reviewed.

Dependencies

Lemma 1.

Source. P. Erdős and M. B. Nathanson, Systems of distinct representatives and minimal bases in additive number theory, in: Number Theory, Carbondale 1979, Lecture Notes in Math. 751, Springer, Berlin, 1979, pp. 89--107 (MR 81k:10089); the edition read is named on the source card.

Bears on

No problem directly. The paper calls it the crucial tool of the paper (p. 96); it drives Theorem 1 and the nonbasis theorems.