Is there a threshold $c$ so every planar set of measure $\ge c$ contains a triangle of area 1? (Erdős #352)
Statement
Is there a constant $c>0$ such that every Lebesgue-measurable set $A\subseteq\mathbb{R}^2$ of measure at least $c$ contains the three vertices of a triangle of area exactly $1$? If such a constant exists, determine the least admissible value of $c$.
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred) that some finite threshold $c$ exists such that every measurable $A\subseteq\mathbb{R}^2$ of measure $\ge c$ contains an area-$1$ triangle — ideally identifying the optimal $c$ (conjectured $4\pi/\sqrt{27}$) — OR a disproof exhibiting, for every $c$, a measurable set of measure $\ge c$ containing no area-$1$ triangle. ADVANCES (each independently checkable): (a) a proof of the conjecture for the union of the interiors of $n$ compact convex sets for some $n\ge 4$, extending the best case stated in the background (Freiling–Mauldin, $1\le n\le 3$), with a rigorous argument (interval-arithmetic-certified if computer-assisted); (b) an improved sufficient threshold — a proof that measure $\ge c'$ forces an area-$1$ triangle for some explicit $c'$, compared against the background values; (c) a proof of the exact-area-$1$ statement (not merely area $>1$) for a class strictly broader than compact convex sets. Deliver the proof file, including any certified computation and an explicit statement of the class and threshold achieved.
Background
Posed by Erdős [Er78d, p.122; Er81b, p.30; Er83d, p.323; Va99, 2.47]; tagged 'geometry' (measure / combinatorial geometry); no prize attached. Frontier: Erdős (unpublished) proved the statement holds if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure — noting it 'follows easily from the Lebesgue density theorem'. He speculated that the optimal threshold is $c=4\pi/\sqrt{27}\approx 2.418$, which would be best possible: a disc of radius just under $2\cdot 3^{-3/4}$ has measure just under $4\pi/\sqrt{27}$ yet contains no triangle of area $1$. Supporting this, Freiling and Mauldin [Ma02] proved that any $A$ with outer measure $>4\pi/\sqrt{27}$ contains the vertices of a triangle of area $>1$, and that the same $4\pi/\sqrt{27}$ threshold holds for the original (area exactly $1$) problem when $A$ is compact and convex. Mauldin [Ma13] further reduced the general problem to the case where $A$ is the union of the interiors of finitely many ($n<\infty$) compact convex sets, and Freiling–Mauldin settled that reduced case for $1\le n\le 3$. A Lean formalisation exists. Listed as open on erdosproblems.com/352 (fetched 2026-07-13, status 'open'). Attacker's tool: attack the finite-union-of-convex-sets reduction — push the Freiling–Mauldin $n\le 3$ result to $n=4$ and beyond via computer-assisted optimisation with interval arithmetic over convex configurations — and numerically certify the extremal-disc bound $4\pi/\sqrt{27}$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #352 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #352 (DeepMind formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.