[#P2680] Largest equilateral-triangle-free subset of the fifteen-row triangular lattice

Problem. Let \(T_{15}=\{(i,j):i\geq0,\ j\geq0,\ i+j<15\}\), embedded with basis vectors meeting at \(60^\circ\). What is the largest subset containing no three vertices of an equilateral triangle?

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1Context

A seeded deletion-and-addition search gives a certified lower bound of 33. The gap to an exact upper bound remains substantial enough for a serious hypergraph search.

2Remarks

Remark 1. All sizes and orientations of equilateral triangles are forbidden.

Remark 2. Squared distance in these coordinates is (i-i')^2+(i-i')(j-j')+(j-j')^2.

3What counts as a solution

Give an equilateral-triangle-free set of maximum size and a complete hypergraph branch certificate excluding the next size.

1Status

What counts as a solution

Current status (The current certified interval is 33 through 56). An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.^[1]

1Packet records

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Record	Kind	Assessment
An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.^[1] Relevance to this problem For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-claim-certified-bounds-33-56 (“The current certified interval is 33 through 56”) records a bound, answer, status fact, or structural consequence. The record states: An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state partial Scope subsets of T_15 with no three vertices of an equilateral triangle in either orientation and at any lattice scale Sources Reference 1 Details Write \(M_{15}\) for the requested maximum. The displayed construction gives \(M_{15}\geq33\). For the upper bound, partition \(T_{15}\) by the two coordinate parities. After division by two, the even-even class is a copy of \(T_8\). Each other class is a copy of \(T_7\). Exhaustive search gives \(\alpha(T_7)=12\). The inner seven rows of \(T_8\) form \(T_7\), and its remaining diagonal has eight vertices, so \(\alpha(T_8)\leq12+8=20\). Every triangle-free subset therefore has size at most \[ 20+3\cdot12=56. \] Thus \(33\leq M_{15}\leq56\). The exact value remains open in this entry. Open full record →
Exact coordinate arithmetic verifies that none of the 2,380 equilateral triangles is contained in the set.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2 Exact coordinate arithmetic verifies that none of the 2,380 equilateral triangles is contained in the set.^[1] Relevance to this problem For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-claim-explicit-33-point-construction (“A 33-point equilateral-triangle-free set”) records a bound, answer, status fact, or structural consequence. The record states: Exact coordinate arithmetic verifies that none of the 2,380 equilateral triangles is contained in the set. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope the stated 33 lattice points, checked against every equilateral triangle among the 120 vertices of T_15 Sources Reference 1 Details A certified construction is \[ \begin{aligned} \{&(0,4),(0,6),(0,7),(0,10),(0,11),(0,14),\\ &(1,4),(1,5),(1,7),(1,9),(1,11),(1,13),\\ &(2,7),(2,8),(2,11),(2,12),(3,2),(3,3),(4,1),(5,0),(6,0),\\ &(7,0),(7,1),(7,2),(8,2),(9,1),(10,0),\\ &(11,0),(11,1),(11,2),(12,2),(13,1),(14,0)\}. \end{aligned} \] The verifier generates every third vertex by applying both 60-degree rotations to each lattice displacement. Deduplication produces 2,380 unordered triples, and none lies wholly in this set. Related: Computation 1 Open full record →
The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.^[1][2] Relevance to this problem For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-attempt-literature-audit (“The nearby literature counts and colors the triangles”) documents a concrete method, search boundary, or failed route. The record states: The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope primary literature on equilateral triangles and forbidden monochromatic equilateral triangles in finite triangular lattices Sources Reference 1, Reference 2 What happened Brouwer, Joe, Noble, and Noble define the same triangular lattice \(T_n\) and the same 3-uniform hypergraph whose edges are equilateral triangles in every orientation. Their formula gives \[ \frac{n^4+2n^3-n^2-2n}{24}=\binom{n+2}{4} \] edges, hence 2,380 at \(n=15\). They determine or bound the minimum number of colors needed to avoid monochromatic equilateral triangles, including \(f(15)\leq5\). Their paper does not report maximum single color-class sizes or \(\alpha(T_{15})\). Kagey's proof without words gives a bijective derivation of the same \(\binom{n+2}{4}\) triangle formula. Searches keyed to the finite triangular lattice, equilateral-triangle-free sets, and hypergraph independence found no primary source settling the 15-row independence number. The certified interval in this entry should therefore be treated as a fresh bounded computation, with novelty still unverified. Related: Computation 1 Open full record →
A standard-library program verifies the 33-point witness and proves alpha(T_7)=12 by exhaustive hypergraph branching.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 A standard-library program verifies the 33-point witness and proves alpha(T_7)=12 by exhaustive hypergraph branching. Relevance to this problem For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-artifact-construction-and-upper-bound (“Executable construction and parity upper-bound certificate”) supplies evidence or a replay used to check the packet. The record states: A standard-library program verifies the 33-point witness and proves alpha(T_7)=12 by exhaustive hypergraph branching. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope exact enumeration on T_15, exact verification of the 33-point witness, and exhaustive exclusion of a 13-point triangle-free subset of T_7 Entry point `Join source_lines with LF characters and execute the resulting Python program` Runtime Python 3.10 or newer, standard library Details The search represents a candidate vertex set by a bit mask. If it contains a triangle, every independent subset must omit at least one vertex of that triangle, so the search branches on its three deletions. A branch with fewer than 13 candidates fails. A triangle-free candidate mask would certify a 13-point set, while memoization merges identical masks. The complete search visits 409,531 calls and 136,510 distinct masks and finds no such set. A checked 12-point witness establishes \(\alpha(T_7)=12\). The same program verifies the 33-point \(T_{15}\) construction, checks the triangle totals, and confirms the four parity-class sizes used in the upper bound. Related: Computation 2 · Computation 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-25. An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56. The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. One 33-point incumbent is listed in the computation record by lattice coordinates.

Computational notes

Exact triple enumeration found 2380 equilateral triangles, agreeing with C(17,4). Five hundred seeded greedy deletion runs found a 33-point set: (0,4),(0,6),(0,7),(0,10),(0,11),(0,14),(1,4),(1,5),(1,7),(1,9),(1,11),(1,13),(2,7),(2,8),(2,11),(2,12),(3,2),(3,3),(4,1),(5,0),(6,0),(7,0),(7,1),(7,2),(8,2),(9,1),(10,0),(11,0),(11,1),(11,2),(12,2),(13,1),(14,0). An exact check found no forbidden triple in this set.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemLargest equilateral-triangle-free subset of the fifteen-row triangular lattice

All 4 recorded relations between these records and the problem

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry
Most squares spanned by twenty points of the ten griddiscrete geometry

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Plain text

“Largest equilateral-triangle-free subset of the fifteen-row triangular lattice.” TheoremDB. P2680. Problem statement; statement text SHA-256 cdde28fdb67570864ed56a4030d866b9d19ac1ab2e14d06b3b4dfdc4ab874bf7. https://theoremdb.org/statement/?ref=P2680

BibTeX

@misc{theoremdb-problem-cdde28fdb67570864ed56a4030d866b9d19ac1ab2e14d06b3b4dfdc4ab874bf7,
  title = {{Largest equilateral-triangle-free subset of the fifteen-row triangular lattice}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 cdde28fdb67570864ed56a4030d866b9d19ac1ab2e14d06b3b4dfdc4ab874bf7},
  url = {https://theoremdb.org/statement/?ref=P2680}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 4 typed links, sourced from arxiv.org^[1], current as of July 25, 2026.

1Lean verification

1References

Packet source. Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, “Problems on the Triangular Lattice”. arXiv:2405.12321 (2024). Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321, abstract and Sections 1-2; Peter Kagey, A Proof Without Words: Triangles in the Triangular Grid, arXiv:2211.00186. ↗preprint · primary source · arXiv:2405.12321, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.The nearby literature counts and colors the triangles. The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.Also cited at Original independence-number target; the cited primary paper studies triangle counts and colorings on the same finite triangular lattice.Also cited at Construction and parity certificate reproduced by tlef15-artifact-construction-and-upper-bound.Also cited at Exact integer-coordinate verification in tlef15-artifact-construction-and-upper-bound.Source used to formulate or check the problem record.For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice: The nearby literature counts and colors the triangles. The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.Source named by the research packet.
Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321, abstract and Sections 1-2; Peter Kagey, A Proof Without Words: Triangles in the Triangular Grid, arXiv:2211.00186. ↗preprint · primary source · arXiv:2211.00186, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.The nearby literature counts and colors the triangles. The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice: The nearby literature counts and colors the triangles. The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.

Original CC0 finite extremal problem with a checked construction.

[#P2680] Largest equilateral-triangle-free subset of the fifteen-row triangular lattice

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Computational notes

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Background

Definitions

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What's known

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Related work

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