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Source. Lemma 5.1, p. 9, of David Turturean, A Negative Answer to Erdős Problem #870, preprint dated April 2026 (11 pp.), https://www.overleaf.com/read/gknkvvxrymfv; the edition read is named on the source card.

Statement

Lemma 5.1 (p. 9). Fix integers $h\ge2$ and $L\ge1$. There are positive integers $N,M$ with $M>N$ such that the finite set $F=[1,N]\cap\mathbb N$ satisfies:

1. for every residue $r$ modulo $M$ there are at least $L$ nondecreasing $h$-tuples $(f_1,\ldots,f_h)$ from $F$ whose sum is congruent to $r$ modulo $M$;
2. there is a residue $\tau$ modulo $M$ such that every sum of at most $h+1$ elements of $F$ that is congruent to $\tau$ modulo $M$ equals $\tau$ as an integer and uses at least $h$ elements of $F$.

Proof pointer

P. 9. With $R$ large in terms of $h$ and $L$, take $N=4R$ and $M=(4h-1)R$. Each residue has a lift between $h$ and $hN$ at distance $\gg_hR$ from both ends, which has $\gg_hR^{h-1}$ representations by $h$-tuples. For $\tau=(4h-2)R$, a sum of at most $h+1$ fillers is at most $4(h+1)R<\tau+M$, so it equals $\tau$, and $\tau>4(h-1)R$ forces at least $h$ fillers.

Dependencies

None. Read depth: claims checked; the statement and proof were read clause by clause on the print.

Bears on

• Problem 870: the deterministic filler gadget of Proposition 5.2, which gives the cases $k\ge4$ of Theorem 1.1. On its own it says nothing about the problem.