# P2870: Consistency strength of projective Ramsey regularity

- ID: `P2870`
- Reference: `projective-sets-ramsey-consistency-strength`
- Page: https://theoremdb.org/statements/P2870
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the exact consistency strength, over \(\mathsf{ZFC}\), of the assertion that every projective subset of \([\omega]^\omega\) is Ramsey. In particular, decide whether this theory is equiconsistent with \(\mathsf{ZFC}\) plus the existence of an inaccessible cardinal.

### Problem setup

- **Definition.** The space \([\omega]^\omega\) consists of all infinite subsets of the natural numbers, identified with their increasing enumerations.
- **Definition.** A set \(A\subseteq[\omega]^\omega\) is Ramsey if some infinite \(H\subseteq\omega\) has either \([H]^\omega\subseteq A\) or \([H]^\omega\cap A=\varnothing\).
- **Remark.** A projective set is obtained from Borel subsets of a Polish space by finitely many applications of continuous image and complementation, equivalently a set in some finite projective level.
- **Definition.** Two theories are equiconsistent if the consistency of either one implies the consistency of the other in the metatheory.

### What counts as a solution

- Prove both relative-consistency directions between \(\mathsf{ZFC}\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength.
- For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answers give upper and lower consistency bounds and say the exact strength is open. Later checked problem lists continue to ask for the missing comparison, but the audit did not locate a definitive current survey. Exact unresolved remainder: Prove both relative-consistency directions between \(\mathsf{ZFC}\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength. For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration. [1](#reference-1) [3](#reference-3) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answers give upper and lower consistency bounds and say the exact strength is open. Later checked problem lists continue to ask for the missing comparison, but the audit did not locate a definitive current survey. Exact unresolved remainder: Prove both relative-consistency directions between \(\mathsf{ZFC}\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength. For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The MathOverflow answers give upper and lower consistency bounds and say the exact strength is open. Later checked problem lists continue to ask for the missing comparison, but the audit did not locate a definitive current survey.

Exact unresolved remainder: Prove both relative-consistency directions between \(\mathsf{ZFC}\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength. For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration.

### Background and intake notes

The target asks for a calibration theorem rather than a new regularity proof alone. Forcing constructions, inner-model lower bounds, and pointclass-by-pointclass implications can be stored with their exact base theories.

- Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow answers give upper and lower consistency bounds and say the exact strength is open. Later checked problem lists continue to ask for the missing comparison, but the audit did not locate a definitive current survey.
- On 2026-07-27 both MathOverflow answers and all comments were checked. They describe a gap between known consistency upper and lower bounds rather than an equiconsistency theorem.
- The assertion quantifies over all finite projective levels at once. Results for a fixed level, for analytic sets, or for lightface definability do not automatically settle it.
- Later open-question notes by Khomskii and set-theory seminar notes were checked for the same Ramsey regularity problem. They continue to record an unresolved consistency-strength gap.
- The base theory and the precise meaning of projective parameters affect consistency calibrations. This record fixes \(\mathsf{ZFC}\) and boldface projective subsets of the standard Polish space \([\omega]^\omega\).
- Trap: Solovay-model results about all sets of reals often use \(\mathsf{ZF}+\mathsf{DC}\) without full choice. They cannot be transferred to the \(\mathsf{ZFC}\) statement without a separate argument.

- Recorded example: Every Borel subset of \([\omega]^\omega\) is Ramsey in \(\mathsf{ZFC}\); the consistency-strength issue arises at higher projective complexity.

### Open directions

- **Route 1** (reported): Prove both relative-consistency directions between \(\mathsf{ZFC}\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength. For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `projective-sets-ramsey-consistency-strength`, 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>MathOverflow question 76280, “Consistency strength of projective Ramsey regularity,” checked 2026-08-01. Original CC0 base-theory formulation written after reading both answers and comments and checking later open-problem notes. https://mathoverflow.net/questions/76280/exact-consistency-strength-of-all-projective-sets-are-ramsey
   - Also cited at Full question, answers, and visible comments concerning Consistency strength of projective Ramsey regularity; checked 2026-08-01.
   - Also cited at Editorial research route recorded 2026-08-01.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Consistency strength of projective Ramsey regularity, the reviewed source scope is Full question, answers, and visible comments concerning Consistency strength of projective Ramsey regularity; checked 2026-08-01.. The packet makes no inference beyond that cited scope.
   - Source named by the research packet.
2. <a id="reference-2"></a>Yurii Khomskii, Giorgio Laguzzi, Benedikt Löwe, and Ilya Sharankou, “Questions on generalised Baire spaces,” Mathematical Logic Quarterly 62(4-5) (2016), 439-456. DOI 10.1002/malq.201600051. abstract and Section 3 questions on regularity in generalized Baire spaces 2^kappa and kappa^kappa https://www.math.uni-hamburg.de/home/khomskii/papers/OpenQ.pdf
   - website; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Consistency strength of projective Ramsey regularity, this source was excluded as status evidence because generalized-Baire questions do not update the classical projective-subset consistency target; it is retained to document source-review history.
3. <a id="reference-3"></a>Stephen G. Simpson, Topics in Logic and Foundations: Spring 2004, graduate lecture notes, revised November 1, 2005. Remark 6.4.12: PRP, analytic sets, the Delta^1_2 counterexample in L, the Solovay-model upper bound, and the stated determinacy gap https://sgslogic.net/t20/notes/topics-s04.pdf
   - website; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Consistency strength of projective Ramsey regularity, this source directly records the classical projective-Ramsey statement, its known bounds, and an open boundary.