Erdős–Shelah Bounds for Union-Free Families

Definitions and mathematical statement

For a positive integer $m$, let $\mathcal F$ be a family of $m$ distinct sets. A subfamily $\mathcal U\subseteq\mathcal F$ is union-free if there are no pairwise distinct $X,Y,Z\in\mathcal U$ such that $X\cup Y=Z$. Define \[ f(m)=\max\left\{k\in\{0,\ldots,m\}:\text{every family of }m\text{ sets contains a union-free subfamily of size at least }k\right\}. \] The supplied statement imposes no restriction on the underlying ground set or on the cardinalities of individual sets.

The assertion attributed in the record to the Erdős--Shelah conjecture is \[ f(m)=(2-o(1))\sqrt m\qquad(m\to\infty), \] so the asserted asymptotic ratio is $f(m)/\sqrt m\to2$.

The record also describes $g(m,a)$ as the largest size guaranteed for an $a$-union-free subfamily and supplies the assertion \[ \lim_{a\to\infty}\liminf_{m\to\infty}\frac{g(m,a)}{\sqrt m}=\infty. \] However, it does not define $a$-union-free or specify the admissible values of $a$. This second display is therefore a preserved source formula, not a fully specified mathematical statement; its forbidden configurations and quantifiers cannot be completed from the supplied evidence.

The registry labels this record open, but also marks its statement unrecoverable. Its supplied verification says that the official report states both displayed claims and immediately records proofs of them, and that no separate residual open problem was recovered. Accordingly, these displays are retained as historical assertions from the extraction, without presenting either as a verified current open problem.

Short English statement

The record concerns guaranteed sizes of union-free subfamilies and attributes an asymptotic bound to Erdős--Shelah. Its proposed generalization lacks a definition of $a$-union-free. Supplied curation reports that both claims were proved in the source report, leaving no recoverable open question; human review is required.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001651$, exact source code catalogue entry, title: Erdős--Shelah Bounds for Union-Free Families. The record belongs to the Oberwolfach Reports Open Problems collection and gives the year 2011; it supplies no individual proposer.
• Supplied source citation: Combinatorics (2011), Oberwolfach Reports, DOI 10.4171/owr/2011/01, pp. 5--83. Source URL: https://doi.org/10.4171/owr/2011/01. The record describes these reports as documentation of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Oberwolfach Report 01/2011, https://doi.org/10.4171/owr/2011/01. The exact supplied literature-check date is 2026-08-21. This assessment is supplied dated curation, not independently verified. It labels the status open while describing the extraction as a report fragment whose displayed Moser result and surrounding theorem are already proved, with no separate residual conjecture stated. The supplied statement-verification field likewise says that both displayed claims are followed by proofs in the report. The background explicitly requests independent verification of time-sensitive status and cited claims.