Transversal theory and matroids

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definitions: Fixes the indexed, labeled-copy, empty-family, and bounded-repetition conventions used throughout the Welsh source unit.

external_inputs: Records the finite Rado theorem and the canonical transversal, Hall, and basis-extension inputs without crediting their proofs to Welsh's paper.

perfect_corollary: Derives the finite rank-defect criterion from Rado's theorem by adjoining free dummy coloops, including all empty and out-of-range cases.

source_corrections: Records two false printed theorems, one missing existence condition, and the lesser notation, endpoint, and index repairs used in this compilation.

theorem_10: Specializes the rank criterion to the loop matroid on a prescribed subset and proves the exact contains-U conditions.

theorem_11: Proves the partition-matroid rank formula and the resulting exact prescribed-multiplicity capacity criterion.

theorem_12: Proves the intersection criterion for two prescribed-multiplicity transversals after establishing both individual feasibility conditions.

theorem_13: Gives a two-point counterexample to the printed common p/k-transversal theorem and isolates the full-rank-versus-base error in its proof.

theorem_13_containment: Proves the containment theorem actually characterized by Welsh's two inequalities, without identifying a full-rank set with a matroid base.

theorem_3: Corrects the omitted feasibility condition and proves the precise relationship between replicated-family transversals and matroid bases.

theorem_4: Proves the rank criterion for an independent p-transversal by applying finite Rado to every indexed copy of the family.

theorem_5: Proves Welsh's two-condition criterion using copied representatives, the labeled-copy rank identity, Perfect's criterion, Hall, and augmentation.

theorem_6: Proves basis exchange for parallel labeled copies and the exact projection rank identity used in the bounded-repetition theorem.

theorem_7: Gives the exact Hall union criterion for a finite prescribed-multiplicity transversal, including zero coordinates and the empty family.

theorem_8: Specializes the bounded-rank criterion to cardinality in the free matroid, retaining the zero and empty-family endpoints.

theorem_9: Gives counterexamples to the printed contains-U statement and explains why its displayed inequality instead belongs to a different theorem.

theorem_9_contained: Proves the criterion actually expressed by Welsh's displayed Theorem 9 inequality, labeled explicitly as a compilation repair.

theorem_9_contains: Proves the full Hall-and-defect criterion for a prescribed-multiplicity transversal whose support contains a fixed set.

transversal_augmentation: Proves that a partial transversal extends as a set to a full transversal whenever the finite indexed family has one.

transversal_rank_formula: States the exact defect-Hall minimum and rank inequalities for the transversal matroid obtained from prescribed family multiplicities.

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D. J. A. Welsh, Transversal Theory and Matroids, Canadian Journal of Mathematics 21 (1969), 1323–1330, DOI 10.4153/CJM-1969-145-0.

The copy read for this card is the eight-page Cambridge published PDF from the official article record; its size is in the source record, with its reading scope and digital-file dates. No distinct manuscript or mathematical version was acquired. The file prints only "https://doi.org/10.4153/CJM-1969-145-0 Published online by Cambridge University Press" on its pages; the journal's article page states "Copyright © Canadian Mathematical Society 1969", offers the article behind a paywall and carries no Creative Commons or open access statement, every other right reserved.

What is reconstructed

The definitions keep repeated family members indexed and replace the source's informal set “of not necessarily distinct elements” (p. 1325) by an indexed representative assignment. The corrected form of Theorem 3 uses replicated family indices to construct the $p$-transversal matroid, with the existence condition omitted from the printed base claim made explicit. The [[set_systems/welsh_1969_transversal_theory_matroids/theorem_4|independent $p$-transversal criterion]] is proved relative to Rado's finite independent-representative theorem.

For bounded repetition, the labeled-copy matroid is proved by basis exchange, including its exact rank formula. The Perfect rank-defect corollary and partial-transversal augmentation are expanded at the precise interfaces used in Theorem 5. The free-matroid specializations give Theorem 7 and Theorem 8.

The source's valid application chain continues through the [[set_systems/welsh_1969_transversal_theory_matroids/theorem_10|contains-$U$ criterion for $k$-transversals]], the partition-capacity criterion, the replicated-transversal rank formula, and the [[set_systems/welsh_1969_transversal_theory_matroids/theorem_12|common $p$/$q$ support theorem]]. These give complete relative reconstructions of the nine valid printed Theorem 3–8 and 10–12 chains.

Two false printed statements

Theorem 9 is false as printed: its prose asks for a $p$-transversal containing $U$, while its displayed inequality is the criterion for one contained in $U$. Neither reading repairs both the prose and formula. Separate compilation results prove the valid [[set_systems/welsh_1969_transversal_theory_matroids/theorem_9_contained|contained-in-$U$]] and [[set_systems/welsh_1969_transversal_theory_matroids/theorem_9_contains|contains-$U$]] criteria.

Theorem 13 is also false at its printed exact-common-support scope. Its proof obtains a $k$-transversal of full rank in the $p$-transversal matroid, which need only contain a $p$-transversal base. The inequalities do characterize that containment conclusion, proved separately in the corrected containment theorem. The counterexamples and all lesser typographical repairs are collected in source corrections. No published erratum or claim that the published theorem itself is wrong is attributed to Welsh; these are explicit findings of this compilation.

Proof boundary

The external-input record keeps Rado's 1942 finite independent-representative theorem external. Its statement is exact, but its original proof has not been compiled here. The ordinary transversal-matroid theorem, Hall's theorem, and finite basis extension link to complete canonical proofs. The Perfect corollary is proved locally relative to Rado. No unrestricted infinite-family theorem, modern status claim, formalization, or Erdős-problem resolution is asserted.

Bears on. No Erdős problem: the paper names none, no problem page cites it, and none of its results is recorded here as bearing on one.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.