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

- ID: `P2680`
- Reference: `triangular-lattice-15-equilateral-free`
- Page: https://theoremdb.org/statements/P2680
- Record maturity: Reviewed problem with recorded work

## 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?

### Remarks

- **Remark.** All sizes and orientations of equilateral triangles are forbidden.
- **Remark.** Squared distance in these coordinates is (i-i')^2+(i-i')(j-j')+(j-j')^2.

### What counts as a solution

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

## Status

An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56. [2](#reference-2)

## Work

### Evidence for the current status

**Computation 1 (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.

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.

### Background and intake notes

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.

- Original intake status: A primary-source search on 2026-07-24 found work on equilateral-triangle colorings of T_n, without an exact independence number for T_15. Novelty remains unverified.
- The triangle hypergraph has 120 vertices and 2380 hyperedges. Branching on a triangle with the fewest available escape vertices gives a reproducible maximum-independent-set search.
- A three-coloring argument supplies a color class lower bound, not an upper bound for one triangle-free set. Confusing those directions is a common trap.
- Symmetry under the dihedral group of the large triangle can canonicalize the first several choices.

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

### Other known results

- **Computation 2** (reproduced): Exact coordinate arithmetic verifies that none of the 2,380 equilateral triangles is contained in the set. [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15. [2](#reference-2) [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library program verifies the 33-point witness and proves alpha(T_7)=12 by exhaustive hypergraph branching.

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `triangular-lattice-15-equilateral-free`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Peter Kagey, “A Proof Without Words: Triangles in the Triangular Grid”. arXiv:2211.00186 (2022). 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 https://arxiv.org/abs/2211.00186
   - preprint; reference source; arXiv:2211.00186, version checked 2026-07-25; checked 2026-07-25
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/2405.12321
   - 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
   - preprint; primary source; arXiv:2405.12321, version checked 2026-07-25; checked 2026-08-01
   - Source use: original_summary
   - 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.