# P2788: Components of the Pasch-switch graph on STS(15) classes

- ID: `P2788`
- Reference: `sts15-pasch-switch-graph`
- Page: https://theoremdb.org/statements/P2788
- Record maturity: Reviewed problem with recorded work

## Problem

Form a graph whose vertices are the \(80\) isomorphism classes of Steiner triple systems on 15 points. Join two distinct classes when labeled representatives differ by one Pasch switch. Determine all connected components and the diameter of each component.

### Problem setup

- **Definition.** A Steiner triple system STS(15) is a family of triples on 15 points in which every pair occurs in exactly one triple.
- **Definition.** A Pasch switch replaces \(abc,ade,fbd,fce\) by \(abd,ace,fbc,fde\) on six distinct points; the two four-block families cover the same pairs.
- **Convention.** Two systems are identified when a permutation of the 15 points maps one block family to the other.

### What counts as a solution

- Provide canonical representatives for all 80 classes, replay every Pasch-switch adjacency, list the connected components, and certify each reported diameter with paths and matching distance lower bounds.

## Status

OPEN (partially resolved): The Pasch-switch graph has two known connected components. One contains 79 isomorphism classes, and the unique anti-Pasch STS(15) class is an isolated vertex. The diameter of the 79-vertex component was not located in the 2026-07-31 literature audit and is the sole remaining question. Provide canonical representatives for all 80 classes, replay every Pasch-switch adjacency, list the connected components, and certify each reported diameter with paths and matching distance lower bounds. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** OPEN (partially resolved): The Pasch-switch graph has two known connected components. One contains 79 isomorphism classes, and the unique anti-Pasch STS(15) class is an isolated vertex. The diameter of the 79-vertex component was not located in the 2026-07-31 literature audit and is the sole remaining question. Provide canonical representatives for all 80 classes, replay every Pasch-switch adjacency, list the connected components, and certify each reported diameter with paths and matching distance lower bounds.

OPEN (partially resolved): The Pasch-switch graph has two known connected components. One contains 79 isomorphism classes, and the unique anti-Pasch STS(15) class is an isolated vertex. The diameter of the 79-vertex component was not located in the 2026-07-31 literature audit and is the sole remaining question.

A complete resolution must satisfy this condition: Provide canonical representatives for all 80 classes, replay every Pasch-switch adjacency, list the connected components, and certify each reported diameter with paths and matching distance lower bounds.

### Background and intake notes

Trade graphs organize local transformations between designs. A complete edge list remains useful for sampling, canonical augmentation, and testing broader trade sets even after the requested invariants are known.

- Original intake status: UNKNOWN: The component and diameter data for this exact quotient graph were not found in the 2026-07-27 search.
- 2026-07-27: Searches found the classification into 80 isomorphism classes, the existence of an anti-Pasch class, and general work on Steiner trades. No complete component and diameter table for this quotient graph surfaced.
- 2026-07-27: The target was checked against the earlier candidate corpora and live prospecting set with no duplicate.
- A useful enumeration should publish canonical block lists, canonical-label hashes, switch endpoints, and shortest-path certificates between eccentric pairs.

- Recorded example: The two four-block families in the definition give one local switch. Any STS containing the first family can be changed by replacing it with the second.

### Open directions

- **Route 1** (reported): Provide canonical representatives for all 80 classes, replay every Pasch-switch adjacency, list the connected components, and certify each reported diameter with paths and matching distance lower bounds. [1](#reference-1)

### Computational notes

- Every STS(15) has \(15\cdot14/6=35\) blocks. No component enumeration was performed for this record.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `sts15-pasch-switch-graph`, 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>F. N. Cole, Louise D. Cummings, and Henry S. White, The Complete Enumeration of Triad Systems in 15 Elements, Proceedings of the National Academy of Sciences 3(3) (1917), 197-199. Complete enumeration of the isomorphism classes on fifteen points. https://doi.org/10.1073/pnas.3.3.197
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - proceedings_article; reference source; checked 2026-08-01
   - Source use: citation_only
   - Primary source for the fact that the quotient graph has exactly 80 vertices.
   - For Components of the Pasch-switch graph on STS(15) classes: Primary source for the fact that the quotient graph has exactly 80 vertices.
   - Source named by the research packet.
2. <a id="reference-2"></a>Rudolf A. Mathon, Kevin T. Phelps, and Alexander Rosa, Small Steiner Triple Systems and Their Properties, Ars Combinatoria 15 (1983), 3-110. Catalogue and standard numbering of the 80 STS(15) classes. https://combinatorialpress.com/ars/vol15/
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Provides the canonical representatives and class invariants needed to construct and audit every vertex of the quotient graph.
   - For Components of the Pasch-switch graph on STS(15) classes: Provides the canonical representatives and class invariants needed to construct and audit every vertex of the quotient graph.
3. <a id="reference-3"></a>Peter B. Gibbons, Computing Techniques for the Construction and Analysis of Block Designs, Ph.D. thesis, University of Toronto, Technical Report 92, 1976. Computational analysis of the STS(15) catalogue under four-cycle, or Pasch, trades. https://library-archives.canada.ca/eng/services/services-libraries/theses/Pages/item.aspx?idNumber=15826048
   - thesis; reference source; checked 2026-08-01
   - Source use: citation_only
   - Primary computational source for the result that 79 of the 80 classes lie in one Pasch-switch component.
   - For Components of the Pasch-switch graph on STS(15) classes: Primary computational source for the result that 79 of the 80 classes lie in one Pasch-switch component.
4. <a id="reference-4"></a>M. J. Grannell, T. S. Griggs, and J. P. Murphy, Switching Cycles in Steiner Triple Systems, Utilitas Mathematica 56 (1999), 3-21. Cycle-switching definitions and the four-cycle case. https://combinatorialpress.com/um/vol56/
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Develops the switching framework in which a Pasch switch is the shortest cycle trade and relates it to transformations among isomorphism classes.
   - For Components of the Pasch-switch graph on STS(15) classes: Develops the switching framework in which a Pasch switch is the shortest cycle trade and relates it to transformations among isomorphism classes.
5. <a id="reference-5"></a>Charles J. Colbourn, Anthony D. Forbes, Mike J. Grannell, Terry S. Griggs, Petteri Kaski, Patric R. J. Östergård, David A. Pike, and Olli Pottonen, Properties of the Steiner Triple Systems of Order 19, Electronic Journal of Combinatorics 17(1) (2010), R98. Section 2.3, especially pages 6-7. https://doi.org/10.37236/370
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - States explicitly that one STS(15) class is anti-Pasch and the other 79 are mutually reachable by Pasch switches, which determines the two connected components.
   - For Components of the Pasch-switch graph on STS(15) classes: States explicitly that one STS(15) class is anti-Pasch and the other 79 are mutually reachable by Pasch switches, which determines the two connected components.
6. <a id="reference-6"></a>Mikhail Klin, Sven Reichard, and Andrew Woldar, Siamese Combinatorial Objects via Computer Algebra Experimentation, in Algorithmic Algebraic Combinatorics and Gröbner Bases, Springer, 2009, 67-112. Section 6.3, A Few Words About STS(15). https://doi.org/10.1007/978-3-642-01960-9_2
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Describes this exact graph on the 80 isomorphism classes and records its component sizes as 79 and 1, identifying the isolated class as STS(15) number 80.
   - For Components of the Pasch-switch graph on STS(15) classes: Describes this exact graph on the 80 isomorphism classes and records its component sizes as 79 and 1, identifying the isolated class as STS(15) number 80.
7. <a id="reference-7"></a>M. J. Grannell and T. S. Griggs, The Pasch Configuration, Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002, 299-300. Definition of the switch and the STS(15) paragraph. https://grannell.net/Papers/Encyc.pdf
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Concise reference for the local move, the unique anti-Pasch class, and connectivity of the remaining 79 classes.
   - For Components of the Pasch-switch graph on STS(15) classes: Concise reference for the local move, the unique anti-Pasch class, and connectivity of the remaining 79 classes.