Keeping every triangle larger than the predicted scale

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Family 191 · Combinatorics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Place $n$ points in a square of side one. You would like every triangle formed by three of them to have reasonably large area. How well can you do?

This is Heilbronn’s triangle problem. It asks about the smallest triangle in a configuration, so one unfortunate triple can spoil an otherwise well-spread arrangement. Keeping points far apart is not sufficient: three widely separated points can lie almost on a line and form a very thin triangle.

The original conjecture predicted that every configuration would contain a triangle of area at most a constant times $n^{-2}$. That prediction was disproved decades ago by constructions improving the scale by a logarithmic factor. A weaker expectation allowed every arbitrarily small loss in the exponent: perhaps a triangle of area at most $n^{-2+\varepsilon}$, up to a constant depending on $\varepsilon$, would always exist.

The manuscript in this family claims a fixed power improvement in the opposite direction. For one absolute $\eta>0$, it constructs arbitrarily large configurations in which every triangle has area at least a constant times $n^{-2+\eta}$. The exponent gain is extremely small, but fixed. That is enough to contradict the weaker expectation as well.

2. Background and essential definitions

The quantity being optimized

For a set $P$ of points in the unit square $[0,1]^2$, define

$$\Delta(P)=\min_{\{p,q,r\}\subset P}\operatorname{Area}(pqr),$$

where the minimum runs over triples of distinct points. Then define

$$\Delta(n)=\max_{\substack{P\subset[0,1]^2\\|P|=n}}\Delta(P).$$

Thus $\Delta(n)$ is the largest minimum triangle area achievable with $n$ points. A lower bound requires constructing a good configuration. An upper bound must apply to every configuration. The directions can be easy to confuse because the definition combines a maximum and a minimum.

A collinear triple has area zero. A positive lower bound therefore guarantees, among other things, that no three selected points lie on a line.

Area as a determinant

For plane points $p,q,r$, elementary linear algebra gives

$$\operatorname{Area}(pqr)=\frac12\left|\det(q-p,r-p)\right|.$$

The determinant measures the signed area of the parallelogram spanned by the two displacement vectors. Taking its absolute value and dividing by two gives the triangle area.

The construction uses another form of the same identity. An integer column $u=(u_1,u_2,u_3)^{\mathsf T}$, with positive third coordinate, represents the point

$$\pi(u)=\left(\frac{u_1}{u_3},\frac{u_2}{u_3}\right).$$

If three columns form a matrix $A$, then their projected triangle has area

$$\frac{|\det A|}{2A_{31}A_{32}A_{33}}.$$

The entries $A_{3j}$ are the third coordinates. Keeping them in a controlled range turns a lower bound on integer determinants into a geometric area bound.

Logarithmic and power improvements

A factor of $\log n$ grows without bound, but more slowly than $n^\eta$ for every fixed $\eta>0$. Consequently, replacing $n^{-2}$ by $(\log n)n^{-2}$ contradicts a constant-factor bound but does not contradict all bounds of the form $n^{-2+\varepsilon}$. A fixed positive exponent gain crosses that second barrier.

3. Development history and previous partial results

Roth’s 1951 paper records Heilbronn’s original question and an elementary construction associated with Erdős. The construction uses a parabola over a finite field, producing grid points with nonzero triangle determinants before scaling. It reaches a minimum area comparable to $n^{-2}$. Random sampling followed by deleting points in small triangles also reaches that basic scale.

Komlós, Pintz, and Szemerédi obtained the stronger lower bound

$$\Delta(n)\ge c\frac{\log n}{n^2}$$

for a positive constant $c$. Their 1982 paper therefore disproved the original constant-factor conjecture. Their method treats small-area triples as forbidden triples in a hypergraph, then selects a large subset containing none. A hypergraph generalizes a graph by allowing an edge to contain more than two vertices. The original paper develops this lower-bound approach.

Upper bounds followed a separate trajectory. Roth obtained the first improvement over the elementary inverse-linear scale. Later work by Schmidt, Roth, and Komlós–Pintz–Szemerédi sharpened the estimates, the last reaching exponents arbitrarily close to $-8/7$.

Cohen, Pohoata, and Zakharov introduced further incidence-geometric methods. Their 2023 work passed the $-8/7$ exponent by a fixed amount, and their subsequent Lower bounds for incidences gives

$$\Delta(n)\le C_\varepsilon n^{-7/6+\varepsilon}$$

for every fixed positive $\varepsilon$. An incidence records a point lying on, or suitably near, a geometric object such as a line; controlling many such relationships can force a small triangle. Their paper develops the incidence estimates and the triangle consequence.

These upper and lower bounds remain widely separated. M1 addresses the lower-bound side and claims to disprove the almost-$n^{-2}$ formulation. It does not close the overall exponent gap or determine the correct asymptotic order of $\Delta(n)$.

4. What the new manuscripts claim

The family contains one manuscript, M1. It states that there are absolute constants $\eta,c_1>0$ and an integer $n_0$ such that

$$\Delta(n)\ge c_1n^{-2+\eta} \qquad\text{for every integer }n\ge n_0.$$

The constants are independent of the number of points. In particular, the improvement is available for every sufficiently large cardinality, not merely for a special sequence.

The weaker conjectural upper bound would say that for every $\varepsilon>0$ there is a constant $C_\varepsilon$ with

$$\Delta(n)\le C_\varepsilon n^{-2+\varepsilon}$$

for all sufficiently large $n$. Taking $\varepsilon=\eta/2$ makes the ratio of M1’s lower bound to the proposed upper bound grow like a positive constant times $n^{\eta/2}$. Thus the two statements cannot both hold for large $n$.

The manuscript gives an explicit admissible expression for $\eta$ in terms of its fixed construction parameters, while emphasizing that it is extremely small and unoptimized. The claim is a qualitative change in the exponent, not a practically large numerical improvement for ordinary-sized point sets.

5. Why the advance matters

Importance: 4/5. The claimed result would overturn a persistent exponent-level expectation in a classical problem of discrete geometry. It also introduces a combination of arithmetic restrictions and random selection that goes beyond the earlier logarithmic improvement.

A tiny fixed exponent matters asymptotically because it eventually exceeds every logarithmic factor. The result would therefore change the plausible long-term scale of the extremal quantity, even if its explicit constants make the construction unsuitable for small examples.

The method connects geometric area to integer determinants, finite-field algebra, and congruences. This suggests a useful way of looking at the problem: instead of distributing points uniformly and then searching for bad triples, build arithmetic restrictions that make most bad triples impossible before sampling.

The rating reflects the conjectural barrier being crossed. It does not treat the claim as a complete solution of Heilbronn’s problem, since the optimal exponent remains undetermined.

6. The main idea

Build rational points through integer columns

The paper starts with integer columns satisfying

$$0\le u_1,u_2<N,\qquad N\le u_3<2N.$$

Their projected points lie in the unit square. If every retained triple has determinant larger than a threshold $\tau$ in absolute value, the area identity gives

$$\operatorname{Area}(pqr)>\frac{\tau}{16N^3}.$$

The task is therefore to sample many columns while making small determinants rare. Zero determinants need particular care because they represent collinear projected triples.

Use finite-field labels to obstruct small nonzero determinants

A finite field is a finite number system in which addition, multiplication, and division by nonzero elements obey the usual field rules. For distinct field elements $x,y,z$, the three columns

$$(1,x,x^2)^{\mathsf T},\quad (1,y,y^2)^{\mathsf T},\quad (1,z,z^2)^{\mathsf T}$$

have nonzero determinant. This is the algebraic reason that a finite-field parabola avoids collinear triples.

M1 assigns random labels in a fixed-degree field extension. A field norm, a multiplicative map from the extension to its smaller base field, packages the nonvanishing determinant information into a polynomial over the base field.

The construction then encodes this polynomial in a selected digit of a much larger integer determinant. A sufficiently large base controls carries between digits. Distinct labels thereby obstruct a small determinant modulo a prescribed modulus. Repeated labels are exceptional events whose probabilities can be made small.

A shared random invertible change of coordinates mixes the residue columns while preserving their determinant. The manuscript combines this with lattice counts to estimate how many integer lifts can have a given small nonzero determinant. A lattice is a discrete set of integer linear combinations of basis vectors; here it organizes the possible lifts satisfying the congruences.

Treat collinearity separately

The nonzero-determinant estimates do not automatically control determinant zero. The latter allows many columns to lie in a common plane before projection, so it requires a separate restriction.

At another prime modulus, M1 uses residues from a small part of a quadratic surface with no three points on a line. Random transformations provide enough flexibility for counting, while the chosen portion excludes certain short integer relations. The Chinese remainder theorem combines the restrictions at the two coprime moduli into compatible integer coordinates.

This second layer supplies the missing control of collinear triples, including degenerate residue patterns. Avoiding those triples is essential because even one would make the minimum area zero.

Delete the few exceptional choices

The final step is an alteration argument: sample more columns than ultimately needed, count repeated projected points and small-determinant triples, and delete one point from each offending pair or triple.

The manuscript’s estimates make the expected number of deletions smaller than the available surplus. Hence a retained set has no bad triples and still contains many points. Relating that count to the determinant threshold gives the fixed exponent gain.

The construction first produces certain cardinalities associated with growing primes. Bounds on the spacing of those sizes then permit passage to every sufficiently large $n$: retain any $n$ points from a suitably sized configuration. Deleting points cannot introduce a smaller triangle.

7. Scope, limitations, and open questions

The theorem concerns ordinary Euclidean triangle area in the unit square. Changing the region by a fixed scaling changes areas by a constant factor, but higher-dimensional simplex questions are not covered by its statement.

All triples must satisfy the bound, including nearly collinear ones. The result is stronger than a statement about average area or about most triples. At the same time, it says nothing about maximizing pairwise distances or controlling the shape of each triangle.

The exponent is fixed but extremely small, and the constants and size threshold are not optimized. The paper’s random construction establishes existence; the stated theorem is not an efficient numerical recipe for optimal configurations at a requested modest size.

The principal remaining problem is the correct decay rate of $\Delta(n)$. The claimed lower bound still lies far below the cited upper bounds. Improving either side, or simplifying the arithmetic mechanism enough to obtain a meaningful exponent, would provide further progress.

8. Sources and further reading

• Roth, On a problem of Heilbronn, Journal of the London Mathematical Society 26 (1951), 198–204, for the early formulation and finite-field construction.
• Komlós, Pintz, and Szemerédi, A lower bound for Heilbronn’s problem, for the logarithmic improvement that disproved the original conjecture.
• Cohen, Pohoata, and Zakharov, A new upper bound for the Heilbronn triangle problem and Lower bounds for incidences, for the recent upper-bound developments.

M1 gives the determinant construction, the separate treatment of zero determinants, and the stated fixed power improvement.

Source manuscripts

• M1. OpenAI, A power improvement in the Heilbronn triangle lower bound (September 25, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.