Add Erdős Problem 593 (obligatory 3-uniform subhypergraphs, $500 prize) by henrykmichalewski · Pull Request #3774 · google-deepmind/formal-conjectures

henrykmichalewski · 2026-04-16T12:23:38Z

fixes #813

Problem

Erdős Problem 593: https://www.erdosproblems.com/593 ($500 prize)

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number > ℵ₀.

…s ($500 prize) Adds formalization of Erdős Problem 593 (Erdős-Galvin-Hajnal, $500 prize). Reference: https://www.erdosproblems.com/593 Characterize finite 3-uniform hypergraphs appearing in every 3-uniform hypergraph of chromatic number > ℵ₀. Includes ThreeUniformHypergraph structure, chromaticCardinal for hypergraphs, 4 fully proved lemmas, and graph analogue variants. Assisted by Claude (Anthropic).

henrykmichalewski · 2026-04-17T08:23:38Z

Closes #813

mo271

Thanks!
There are some problems here, see comments, I only reviewed up to the first trivially provable open (or not so open) statement

mo271 · 2026-04-17T09:11:55Z

+
+**Problem (Erdős, $500)**: Characterize those finite 3-uniform hypergraphs which appear in
+every 3-uniform hypergraph of chromatic number $> \aleph_0$.
+
+**Background:** A hypergraph $H = (V, E)$ is **$r$-uniform** if every hyperedge $e \in E$
+has exactly $r$ vertices. The **chromatic number** $\chi(H)$ of a hypergraph is the minimum
+number of colors needed to color its vertices so that no hyperedge is monochromatic. A finite
+$r$-uniform hypergraph $F$ is **obligatory** (for the class of $r$-uniform hypergraphs with
+chromatic number $> \aleph_0$) if every $r$-uniform hypergraph with chromatic number
+$> \aleph_0$ contains a copy of $F$ as a sub-hypergraph.
+
+**Known (graph case, $r = 2$, Erdős–Galvin–Hajnal [EGH75]):** For graphs (2-uniform
+hypergraphs), the problem is completely solved:
+- A graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs.
+- No fixed odd cycle is obligatory: for every odd $k$, there exists a graph with chromatic
+  number $\aleph_1$ that contains no cycle of length $k$.
+
+The 3-uniform case remains **open**.
+
+**Formalization notes:** We represent a 3-uniform hypergraph on vertex type `V` as a pair
+`(edges, uniform)` where `edges : Set (Finset V)` and every edge has cardinality 3. A proper
+coloring sends vertices to colors such that no hyperedge is monochromatic. The chromatic
+cardinal is the infimum of cardinalities of color types admitting a proper coloring. A finite
+hypergraph `F` *appears* in `H` if there is an injective vertex map carrying edges of `F`
+into edges of `H`.
+
+We work at universe level `Type` (universe 0) throughout to avoid universe metavariable issues.


All this stuff here should go to where it belongs, for instance the Formalisation notes on 3-uniform hypergraphs closer to where the hypergraphs are defined

mo271 · 2026-04-17T09:13:18Z

+/- ## Definitions for 3-uniform hypergraphs -/
+
+/-- A **3-uniform hypergraph** on vertex type `V` is a set of 3-element `Finset`s.
+Each element of `edges` is a hyperedge, and `uniform` ensures each has exactly 3 vertices. -/
+structure ThreeUniformHypergraph (V : Type) where
+  /-- The set of hyperedges: each edge is a 3-element finset of vertices. -/
+  edges : Set (Finset V)
+  /-- Every hyperedge has exactly 3 vertices. -/
+  uniform : ∀ e ∈ edges, e.card = 3
+
+/-- A **proper coloring** of a 3-uniform hypergraph `H` by a color type `C` is a vertex
+coloring such that no hyperedge is monochromatic (all three vertices receive the same color). -/
+def ThreeUniformHypergraph.IsProperColoring {V : Type} (H : ThreeUniformHypergraph V)
+    {C : Type} (f : V → C) : Prop :=
+  ∀ e ∈ H.edges, ∃ u ∈ e, ∃ v ∈ e, f u ≠ f v
+
+/-- The **chromatic cardinal** of a 3-uniform hypergraph `H` is the infimum of cardinalities
+of color types admitting a proper coloring. We use `Cardinal.{0}` matching `Type`. -/
+noncomputable def ThreeUniformHypergraph.chromaticCardinal {V : Type}
+    (H : ThreeUniformHypergraph V) : Cardinal.{0} :=
+  sInf {κ : Cardinal.{0} | ∃ (C : Type), #C = κ ∧ ∃ f : V → C, H.IsProperColoring f}
+
+/-- A finite 3-uniform hypergraph `F` **appears** in `H` (as a sub-hypergraph) if there
+exists an injective vertex map `φ : W → V` that sends every hyperedge of `F` to a hyperedge
+of `H`. -/
+def ThreeUniformHypergraph.Appears {W V : Type} [DecidableEq V]
+    (F : ThreeUniformHypergraph W) (H : ThreeUniformHypergraph V) : Prop :=
+  ∃ φ : W → V, Function.Injective φ ∧
+    ∀ e ∈ F.edges, e.image φ ∈ H.edges
+
+/-- A finite 3-uniform hypergraph `F` on a `Fintype` vertex type is **obligatory** if it
+appears in every 3-uniform hypergraph (on a `Type`-valued vertex set) whose chromatic
+cardinal exceeds `ℵ₀`. -/
+def IsObligatory {W : Type} [Fintype W] (F : ThreeUniformHypergraph W) : Prop :=
+  ∀ (V : Type) [DecidableEq V] (H : ThreeUniformHypergraph V),
+    ℵ₀ < H.chromaticCardinal → F.Appears H


Those definitions should go in our FormalConjecturesForMathlib dir in an appropriate place. I'm surprised that Mathlib doesn't have hypergraphs already

mo271 · 2026-04-17T09:16:17Z

+theorem erdos_593 : answer(sorry) ↔
+    ∃ (P : ∀ (W : Type) [Fintype W], ThreeUniformHypergraph W → Prop),
+      ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),
+        IsObligatory F ↔ P W F := by
+  sorry


AS stated this is trivially true:

theorem erdos_593 : answer(True) ↔ ∃ (P : ∀ (W : Type) [Fintype W], ThreeUniformHypergraph W → Prop), ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F ↔ P W F := by exact ⟨fun _ => ⟨fun _ _ F => IsObligatory F, fun _ _ _ => Iff.rfl⟩, fun _ => trivial⟩

The original problem asks to characterize which finite 3-uniform hypergraphs are obligatory i.e., to give a structural description in terms of simpler properties (analogous to "bipartite" in the graph case). Bare existential quantification over predicates cannot capture this: in formal logic, every property trivially characterizes itself.

This is a known difficulty with formalizing "characterize X" problems. Potentially our answer(sorry) mechanism can help here?

Good catch — applied your suggestion in 6565a7f (April 17, same day as the review). The trivial existential pattern is gone; the file now states two substantive answer(sorry) theorems capturing the open conjecture from each direction:

@[category research open, AMS 5] theorem erdos_593 : answer(sorry) ↔ ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F → F.IsTwoColorable @[category research open, AMS 5] theorem erdos_593.variants.two_colorable_implies_obligatory : answer(sorry) ↔ ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W), F.IsTwoColorable → IsObligatory F

The conjecture is that IsObligatory F ↔ F.IsTwoColorable (analogous to the Erdős–Galvin–Hajnal characterisation of obligatory graphs as the bipartite ones); we file each direction as its own open question via answer(sorry) so neither is trivially provable. Hopefully this captures what you suggested with the answer(sorry) mechanism.

… fix trivial statement Per mo271's review feedback on PR google-deepmind#3774 (Problem 593, $500 prize): 1. Refactor: move ThreeUniformHypergraph, IsProperColoring, chromaticCardinal, Appears, IsTwoColorable, and IsObligatory out of the problem file into FormalConjecturesForMathlib/Combinatorics/Hypergraph/ThreeUniform.lean and register it in FormalConjecturesForMathlib.lean. 2. Fix the trivially-true main theorem: the old statement '∃ P, ∀ F, IsObligatory F ↔ P W F' was proved by the reviewer with a one-liner (P := IsObligatory). Replace with two non-trivial open conjectures based on the 2-colorability (Property B) characterization: - erdos_593: every obligatory finite 3-uniform hypergraph is 2-colorable - erdos_593.variants.two_colorable_implies_obligatory: the converse These mirror the graph case (EGH75: obligatory graphs = bipartite graphs). Co-Authored-By: Claude Opus 4 (1M context) <noreply@anthropic.com>

… fix trivial statement Addresses review feedback by mo271: - Move ThreeUniformHypergraph + related definitions to FormalConjecturesForMathlib/Combinatorics/Hypergraph/ThreeUniform.lean - Import them from new location in 593.lean - Replace trivially-true main theorem with two substantive open conjectures: - erdos_593 (necessary condition): obligatory ⟹ 2-colorable - variants.two_colorable_implies_obligatory (converse, open) Assisted by Claude (Anthropic).

Mirrors the Round C docstring pass from the private repo's phase1-infrastructure branch. Each Lean file now carries the canonical source statement and upstream URL inline so reviewers can verify formalization against the source without navigating away from the diff.

henrykmichalewski · 2026-05-05T11:03:31Z

@mo271 — back here after a long pause. All three of your April 17 review comments were addressed in 6565a7f the same day. Recap, since the PR's been quiet:

Your comment	Status
L57: "all this stuff here should go to where it belongs" (formalisation notes)	Moved closer to the `ThreeUniformHypergraph` definitions in the new infra module. GitHub marks the thread outdated.
L99: "those definitions should go in `FormalConjecturesForMathlib`"	Moved to `FormalConjecturesForMathlib/Combinatorics/Hypergraph/ThreeUniform.lean` (105 lines: `ThreeUniformHypergraph`, `IsProperColoring`, `chromaticCardinal`, `Appears`, `IsTwoColorable`, `IsObligatory`). 593.lean now just imports them.
L115: "trivially true as stated"	Replaced the `answer(True) ↔ ∃ P, ...` pattern with two `answer(sorry)` theorems stating the necessary direction (`IsObligatory F → F.IsTwoColorable`) and its converse, per your `answer(sorry)` suggestion. Reply on the inline thread.

PR is MERGEABLE (CLA + CI green); ready for re-review whenever you have a moment.

Paul-Lez · 2026-05-07T17:06:29Z

+**Prize:** \$500 (see erdosproblems.com/593).
+-/
+@[category research open, AMS 5]
+theorem erdos_593 : answer(sorry) ↔


Does the source ask for a characterization of the finite 3-uniform hypergraphs that are obligatory, rather than just the necessary direction IsObligatory F → F.IsTwoColorable? I think it would be safer to either state the characterization predicate directly, with the unknown class behind answer(sorry), or make this an iff if 2-colorability is intended as the conjectured characterization.

Paul-Lez · 2026-05-07T17:14:01Z

+numbers as colors gives a proper coloring, so $\chi(H) \leq \#\mathbb{N} = \aleph_0$,
+contradicting $\chi(H) > \aleph_0$.
+-/
+@[category undergraduate, AMS 5]


nit: Looks like this file still uses @[category undergraduate] for a few auxiliary/supporting lemmas. This now no longer works on main - the new version of this is textbook.

…ul-Lez review) Per Paul-Lez review on PR google-deepmind#3774, line 94: the source asks for a characterization of the obligatory finite 3-uniform hypergraphs, not just the necessary direction `IsObligatory F → F.IsTwoColorable`. The natural conjectured characterization (analogous to Erdős–Galvin–Hajnal for graphs: bipartite ⇔ obligatory) is IsObligatory F ↔ F.IsTwoColorable. Rewrote the headline `erdos_593` as this iff, behind `answer(sorry)`. The two directions are kept as separate `answer(sorry)`-d open variants: - obligatory_implies_two_colorable (necessary direction; new, replaces the previous one-direction headline) - two_colorable_implies_obligatory (sufficient direction; pre-existing) Note: Paul-Lez's separate `category undergraduate` → `category textbook` rename nit is deferred to a follow-up — the rename would require merging upstream/main into this branch first to pull in the new CategoryLinter keyword (the branch-local linter still expects `undergraduate`). Builds cleanly: lake build FormalConjectures.ErdosProblems.«593»

henrykmichalewski · 2026-05-08T00:48:12Z

@Paul-Lez — addressed your characterization comment in eb09c34. The headline erdos_593 is now stated as the conjectured iff:

@[category research open, AMS 5]
theorem erdos_593 : answer(sorry) ↔
    ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),
      IsObligatory F ↔ F.IsTwoColorable := by
  sorry

with both directions kept as separate answer(sorry)-d open variants (obligatory_implies_two_colorable and the existing two_colorable_implies_obligatory).

Your category undergraduate → category textbook rename nit is deferred to a separate follow-up: the rename requires merging upstream/main into this branch first, since the branch-local CategoryLinter currently rejects textbook. I'll bundle that with the PR #3775 merge-main work and apply it consistently across all our open branches.

Builds cleanly.

mo271

Thanks!
There are a few comments which more widely apply to other formalisations and a few specific to the math here. (It is now mostly good, i.e. I didn't find trivial disproofs and it looks like the formalisation faithfully represents the informal maths...)

mo271 · 2026-05-10T08:10:38Z

+**Verbatim statement (Erdős #593, status O):**
+> Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.
+
+**Source:** https://www.erdosproblems.com/593
+
+**Notes:** OPEN - $500


Suggested change

**Verbatim statement (Erdős #593, status O):**

> Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.

**Source:** https://www.erdosproblems.com/593

**Notes:** OPEN - $500

mo271 · 2026-05-10T08:12:17Z

+*Reference:* [erdosproblems.com/593](https://www.erdosproblems.com/593)
+
+*References for known results:*


Suggested change

*Reference:* [erdosproblems.com/593](https://www.erdosproblems.com/593)

*References for known results:*

*References:*

- [erdosproblems.com/593](https://www.erdosproblems.com/593)

(Seeing this for the third time in your Erdős pull request: we generally just list all references under References and don't distinguis between one main one and references for known results, please fix in all of the pending prs)

mo271 · 2026-05-10T08:15:07Z

+**Problem (Erdős, $500)**: Characterize those finite 3-uniform hypergraphs which appear in
+every 3-uniform hypergraph of chromatic number $> \aleph_0$.
+
+**Background:** A hypergraph $H = (V, E)$ is **$r$-uniform** if every hyperedge $e \in E$
+has exactly $r$ vertices. The **chromatic number** $\chi(H)$ of a hypergraph is the minimum
+number of colors needed to color its vertices so that no hyperedge is monochromatic. A finite
+$r$-uniform hypergraph $F$ is **obligatory** (for the class of $r$-uniform hypergraphs with
+chromatic number $> \aleph_0$) if every $r$-uniform hypergraph with chromatic number
+$> \aleph_0$ contains a copy of $F$ as a sub-hypergraph.
+
+**Known (graph case, $r = 2$, Erdős–Galvin–Hajnal [EGH75]):** For graphs (2-uniform
+hypergraphs), the problem is completely solved:
+- A graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs.
+- No fixed odd cycle is obligatory: for every odd $k$, there exists a graph with chromatic
+  number $\aleph_1$ that contains no cycle of length $k$.
+
+The 3-uniform case remains **open**.
+
+**Formalization notes:** See `FormalConjecturesForMathlib.Combinatorics.Hypergraph.ThreeUniform`
+for the definitions of `ThreeUniformHypergraph`, `IsProperColoring`, `chromaticCardinal`,
+`Appears`, and `IsObligatory` used throughout this file.
+
+Note: Mathlib does not yet have a general hypergraph API; the infrastructure in
+`FormalConjecturesForMathlib.Combinatorics.Hypergraph.ThreeUniform` fills that gap.


Suggested change

**Problem (Erdős, $500)**: Characterize those finite 3-uniform hypergraphs which appear in

every 3-uniform hypergraph of chromatic number $> \aleph_0$.

**Background:** A hypergraph $H = (V, E)$ is **$r$-uniform** if every hyperedge $e \in E$

has exactly $r$ vertices. The **chromatic number** $\chi(H)$ of a hypergraph is the minimum

number of colors needed to color its vertices so that no hyperedge is monochromatic. A finite

$r$-uniform hypergraph $F$ is **obligatory** (for the class of $r$-uniform hypergraphs with

chromatic number $> \aleph_0$) if every $r$-uniform hypergraph with chromatic number

$> \aleph_0$ contains a copy of $F$ as a sub-hypergraph.

**Known (graph case, $r = 2$, Erdős–Galvin–Hajnal [EGH75]):** For graphs (2-uniform

hypergraphs), the problem is completely solved:

- A graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs.

- No fixed odd cycle is obligatory: for every odd $k$, there exists a graph with chromatic

number $\aleph_1$ that contains no cycle of length $k$.

The 3-uniform case remains **open**.

**Formalization notes:** See `FormalConjecturesForMathlib.Combinatorics.Hypergraph.ThreeUniform`

for the definitions of `ThreeUniformHypergraph`, `IsProperColoring`, `chromaticCardinal`,

`Appears`, and `IsObligatory` used throughout this file.

Note: Mathlib does not yet have a general hypergraph API; the infrastructure in

`FormalConjecturesForMathlib.Combinatorics.Hypergraph.ThreeUniform` fills that gap.

Not need to repeat the statement etc in the module docstring: generally the problem statement and variants for the problem should basically only be written down once, i.e. we generally want to have exactly one verbatim copy of what is written on erdosproblems.com spread across the entire file -- statements of variants should go to the lean statement of those variants and not in the module docstring.
Please fix in all pending prs.

mo271 · 2026-05-10T08:16:28Z

+
+**Status:** OPEN.
+
+**Prize:** \$500 (see erdosproblems.com/593).


There is never a need to mention the **Status**: those are captured in the category tag (research open) and should not be duplicated. Also the "price" shouldn't be mentioned more than once.

Suggested change

**Status:** OPEN.

**Prize:** \$500 (see erdosproblems.com/593).

Please also fix in other pending prs.

mo271 · 2026-05-10T08:20:27Z

+
+**Status:** OPEN.


Suggested change

**Status:** OPEN.

mo271 · 2026-05-10T08:20:40Z

+**Status:** OPEN. Together with `erdos_593.variants.obligatory_implies_two_colorable`,
+this implies `erdos_593`.


Suggested change

**Status:** OPEN. Together with `erdos_593.variants.obligatory_implies_two_colorable`,

this implies `erdos_593`.

Together with `erdos_593.variants.obligatory_implies_two_colorable`,

this implies `erdos_593`.

potentially this implication could also be added as formal statement?

mo271 · 2026-05-10T08:21:58Z

+
+**Status:** SOLVED [EGH75].


Suggested change

**Status:** SOLVED [EGH75].

mo271 · 2026-05-10T08:22:08Z

+
+**Status:** SOLVED [EGH75].


Suggested change

**Status:** SOLVED [EGH75].

mo271 · 2026-05-10T12:58:24Z

+    ∀ (V : Type*) (G : SimpleGraph V),
+      ℵ₀ < G.chromaticCardinal →
+      ∀ (W : Type*) [Fintype W] (F : SimpleGraph W), F.IsBipartite →
+        Nonempty (F →g G) := by


graph_case_bipartite_obligatory uses Nonempty (F →g G) (existence of a graph homomorphism) to express "F appears in G", but the hypergraph Appears definition requires an injective vertex map. These are different notions.

Consider using Nonempty (F ↪g G) (graph embedding) or at least document why homomorphism suffices if it does.

Done in 7170485. graph_case_bipartite_obligatory now uses Nonempty (F ↪g G) (graph embedding), aligned with the injective vertex map of the hypergraph Appears definition; the docstring caveat is removed. Build green.

- Collapse the two reference blocks into a single *References:* block in the module docstring; drop the duplicated verbatim statement, background paragraph, and formalization note. - Trim the erdos_593 theorem docstring to the verbatim Erdős statement plus a single sentence about the conjectural characterization. - Drop the **Status:** / **Prize:** mentions on erdos_593 and the three solved/open variants (already captured by @[category]). - Add erdos_593.variants.implications_combine: the elementary Iff derivation showing that the two open implications jointly imply the characterization in erdos_593 (mo271 line-123 follow-up). Follow-up flagged in mo271 line-147 review thread: graph_case_bipartite_obligatory uses Nonempty (F →g G) (graph homomorphism) but the hypergraph Appears definition uses an injective vertex map. The two notions differ; switching to Nonempty (F ↪g G) (graph embedding) is left as a follow-up. A docstring note records the discrepancy so it isn't lost.

henrykmichalewski · 2026-05-10T17:35:22Z

Applied mo271's docstring-convention review pass in 53f4f31:

Lines 28, 33, 64: collapsed the two reference blocks into a single *References:* block; dropped the duplicated verbatim statement and background paragraph from the module docstring.
Lines 89, 105, 139, 158: removed the Status: / Prize: mentions from erdos_593, the obligatory-implies-two-colorable variant, and the two graph-case variants (already captured by @[category]).
Line 123 (substantive follow-up): added erdos_593.variants.implications_combine, the elementary Iff derivation showing that the two open implications jointly imply the conjectured characterization in erdos_593.

Follow-up flagged in the line-147 thread: graph_case_bipartite_obligatory uses Nonempty (F →g G) (graph homomorphism) but the hypergraph Appears definition uses an injective vertex map. Switching to Nonempty (F ↪g G) (graph embedding) is left as a follow-up; a docstring note on the theorem records the discrepancy so it isn't lost.

Build: lake build FormalConjectures.ErdosProblems.«593» is green.

mo271

Regarding:

Note: this statement uses Nonempty (F →g G) (graph homomorphism), which is weaker than the
injective vertex map used in the hypergraph Appears definition. A future revision may switch
to Nonempty (F ↪g G) (graph embedding) to align with the hypergraph notion.

Let's do this now, rather than in a follow-up

mo271 · 2026-05-11T07:28:27Z

+  sorry
+
+/--
+**Erdős Problem 593 — Necessary direction (open)**: Every obligatory finite 3-uniform


Suggested change

**Erdős Problem 593 — Necessary direction (open)**: Every obligatory finite 3-uniform

**Erdős Problem 593 — Necessary direction**: Every obligatory finite 3-uniform

Let's remove the "open" from docstrings altogether -- those are recorded in the category tag and will just be a pain to keep in sync when problems get solved

Applied the suggestion plus the same rule across all docstrings in the file (necessary/sufficient direction headlines and the two graph-case variants no longer carry (open)/(solved) suffixes — those are captured by the @[category] tag). Commit: 7170485.

…edding (mo271 review); drop open/solved from docstrings

henrykmichalewski · 2026-05-12T08:40:11Z

Addressed mo271's two follow-ups in 7170485:

Line 147 (graph homomorphism → embedding): graph_case_bipartite_obligatory now uses Nonempty (F ↪g G) (SimpleGraph.Embedding, Mathlib.Combinatorics.SimpleGraph.Maps), aligned with the injective vertex map of the hypergraph Appears definition; the docstring caveat is removed.
Line 56 + sweep (drop (open)/(solved) from docstrings): applied across all four affected docstrings (necessary/sufficient direction headlines and the two graph-case variants). Status is now solely captured by @[category research open] / @[category research solved].

Build: lake build FormalConjectures.ErdosProblems.«593» is green.

github-actions Bot added the erdos-problems Erdős Problems label Apr 16, 2026

mo271 reviewed Apr 17, 2026

View reviewed changes

henrykmichalewski force-pushed the add-problem-593 branch from 49b3486 to 6565a7f Compare April 19, 2026 09:56

Paul-Lez reviewed May 7, 2026

View reviewed changes

Paul-Lez added the awaiting-author The author should answer a question or perform changes. Reply when done. label May 7, 2026

henrykmichalewski added 2 commits May 8, 2026 13:52

Merge remote-tracking branch 'upstream/main' into add-problem-593

1884966

fix(ErdosProblems/593): @[category undergraduate] → @[category textbo…

a4e2648

…ok] + copyright 2026 Per Paul-Lez review on PR google-deepmind#3774. Mechanical nits applied on top of an upstream/main merge to pick up the new attribute infrastructure (google-deepmind#3900).

mo271 reviewed May 10, 2026

View reviewed changes

mo271 reviewed May 11, 2026

View reviewed changes

fix(ErdosProblems/593): switch graph_case_bipartite_obligatory to emb…

7170485

…edding (mo271 review); drop open/solved from docstrings

		Reference: [erdosproblems.com/593](https://www.erdosproblems.com/593)

		References for known results:


		Status: OPEN.

		Prize: \$500 (see erdosproblems.com/593).

		Status: OPEN. Together with `erdos_593.variants.obligatory_implies_two_colorable`,
		this implies `erdos_593`.

	Erdős Problem 593 — Necessary direction (open): Every obligatory finite 3-uniform
	Erdős Problem 593 — Necessary direction: Every obligatory finite 3-uniform

Conversation

henrykmichalewski commented Apr 16, 2026 • edited by mo271 Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

Problem

Contents

Uh oh!

henrykmichalewski commented Apr 17, 2026

Uh oh!

mo271 left a comment

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

henrykmichalewski commented May 5, 2026

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

henrykmichalewski commented May 8, 2026

Uh oh!

mo271 left a comment

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

henrykmichalewski commented May 10, 2026

Uh oh!

mo271 left a comment

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

henrykmichalewski commented May 12, 2026

Uh oh!

Reviewers

Assignees

Labels

Projects

Milestone

Development

Uh oh!

3 participants

henrykmichalewski commented Apr 16, 2026 •

edited by mo271

Loading