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Math-2418 · mathematics · Combinatorics

Packing and Covering for Infinite Matroids

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $M$ and $N$ be matroids on a common ground set $E$. Does there exist a partition $E=P\mathbin{\dot\cup}C$ such that the restrictions $M|P$ and $N|P$ have disjoint spanning sets, while the contractions $M.C$ and $N.C$ have independent sets whose union is $C$?

Formulation references: [S1], [S2].

Short English statement

Let $M$ and $N$ be matroids on a common ground set $E$. Does there exist a partition $E=P\mathbin{\dot\cup}C$ such that the restrictions $M|P$ and $N|P$ have disjoint spanning sets, while the contractions $M.C$ and $N.C$ have independent sets whose union is $C$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2418-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2418

Unrestricted infinite matroid obstruction: The claimed countable self-dual partitional pair has no packing-covering partition even though the conjecture quantifies over arbitrary matroid pairs and ground sets.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Packing and Covering for Infinite Matroids

OpenAI's claim

Unrestricted infinite matroid obstruction: Two self-dual partitional matroids on a countable common ground set are claimed to admit no witness for intersection and no partition satisfying packing/covering. They lie outside the finitary and cofinitary classes.

Mathematical statement of the claimed result

Mathematical claim 1.
In ZFC the claimed construction gives two matroids $M_0,M_1$ on a countably infinite set $E$, each equal to its dual on that same labelled ground set, for which \[I_0\cup I_1\ne E\quad\text{for every }I_i\text{ independent in }M_i.\] No partition $E=P\mathbin{\dot\cup}C$ has disjoint spanning sets in the restrictions to $P$ and independent sets covering $C$ in the contractions to $C$. The intersection corollary also states that, for every common independent $J$ and partition $J=J_0\mathbin{\dot\cup}J_1$, $\operatorname{cl}_{M_0}(J_0)\cup\operatorname{cl}_{M_1}(J_1)\ne E$.

Outline of the argument

Unrestricted infinite matroid obstruction: An ultrafilter carries an ordinal-valued rank on large subsets. A two-sided self-dual matroid is built from rapidly growing blocks and probe limits so that any relevant basis forces a strict rank decrease from one side to the other. Alternating copies along an infinite bundle path would turn a supposed covering into an infinite descending ordinal chain.

Formal verification

Family 185 describes Lean coverage for the pair of countable self-dual partitional matroids and its covering, packing, and intersection obstructions.

Unrestricted infinite matroid obstruction: The manuscript links InfiniteMatroid.lean (declaration OAI.InfiniteMatroidCounterexample.main), InfiniteMatroidCorollaries.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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