Sarkozi 1965 uber ein problem von erdos und

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satz: The Sárközy-Szemerédi theorem of 1965 that for distinct positive reals 0 < a_1 < ... < a_n and every epsilon > 0, once n exceeds n_0(epsilon) no value t is a subset sum in more than (1+epsilon)(8/sqrt(pi)) 2^n/n^{3/2} ways; the Erdős-Moser conjecture without its logarithmic factor.

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Sárközi, A. and Szemerédi, E., Über ein Problem von Erdös und Moser. Acta Arith. (1965), 205-208. The journal record is Acta Arithmetica 11 (1965), no. 2, 205--208, DOI 10.4064/aa-11-2-205-208 (Crossref); the paper was received by the editors on 26 November 1964 (p. 208). The paper prints the first author as A. Sárközy; the site's key [SaSz65], Erdős's 1973 survey and this folder's slug write Sárközi.

This short German paper (read from a clean scan) proves the conjecture of Erdős and Moser on the maximal number of equal subset sums. For distinct positive reals 0 < a_1 < a_2 < ... < a_n, let f(t) be the number of solutions of sum epsilon_i a_i = t with each epsilon_i in {0,1}. Erdős and Moser had proved max_t f(t) < c_1 2^n n^{-3/2} log^{3/2} n and conjectured the log factor could be removed; the paper's Satz establishes that for every epsilon > 0 and n > n_0(epsilon), max_t f(t) < (1 + epsilon)(8/sqrt(pi)) 2^n n^{-3/2}, which is sharp in order since a_i = i already gives max f(t) > c_3 2^n n^{-3/2}. The proof is indirect: assuming f(t) >= (1 + epsilon)(8/sqrt(pi)) 2^n n^{-3/2} for some t, the solution sets are split according to their intersections with the small elements and the large elements, and a Sperner-type lemma (a modified and weaker form of a theorem of Katona, as the paper says; stated on pp. 205-206 and proved on p. 206 via Sperner's theorem) produces two solution sets whose traces on the two parts are nested, giving a contradiction through the counting inequalities (5)-(11). The result is the bound recorded for Problem 362 on the largest number of subsets of a set of n distinct numbers with the same sum.

Source: https://www.impan.pl/shop/publication/transaction/download/product/95839.

Retained artifact. The folder-name PDF is a three-page scan of two-page spreads without a text layer: PDF p. 1 shows printed p. 204 (the end of the preceding paper) and p. 205, PDF p. 2 printed pp. 206--207, and PDF p. 3 printed p. 208 and the first page of the following paper. Everything on this card was read on the page images (130 dpi renders) on 2026-09-18. The paper's maximum is over all real $t\ge0$ ($\max_{0\le t<+\infty}f(t)$), and the $a_i$ are arbitrary distinct positive reals; its reference [1] is Katona, "On a conjecture of Erdős, and a stronger form of Sperner's theorem", Magyar Tud. Akad. Mat. Kutató Int. Közl., "im Druck", cited both for the Erdős--Moser bound and for the theorem the Lemma modifies, and [2] is Sperner's 1928 paper. No notice is printed on the scan's first or last pages; the publisher's record (https://www.impan.pl/get/doi/10.4064/aa-11-2-205-208, read 2026-10-02) offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site), a Creative Commons Attribution license whose version the record does not name; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Read status: claims checked for the introduction (the definition of $f(t)$, the Erdős--Moser bound, the conjecture and the lower bound for $a_i=i$), the Satz and the Lemma (printed pp. 205--206), read clause by clause on the page images; the proof (pp. 205--208) was read for structure and not checked. Nothing here is independently reviewed.

Bears on. #362 (the Satz, printed p. 205, PDF p. 1, page image: $\max_tf(t)<(1+\varepsilon)(8/\sqrt\pi)2^n/n^{3/2}$ for $n>n_0(\varepsilon)$, the first question's bound; the introduction's report of the Erdős--Moser bound with the factor $\log^{3/2}n$)

Results to transcribe.

• Satz (p. 205): For every epsilon > 0 and n > n_0(epsilon), any distinct positive reals a_1 < ... < a_n satisfy max_t f(t) < (1 + epsilon)(8/sqrt(pi)) 2^n n^{-3/2}, where f(t) counts subsets with sum t. Paged as satz.
• Lemma (pp. 205-206): A modified and weaker form of a theorem of Katona: if A = B union C with B and C disjoint, |B| = b, |C| = c, and M_1, ..., M_l are subsets of A with l >= 2^b binomial(c, floor(c/2)) + 1, then some M_u, M_v satisfy M_u ∩ B = M_v ∩ B and M_u ∩ C contained in M_v ∩ C.