Author:
Jonathan David Farley Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, United States of America

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Problem 2 of Welsh’s 1976 text Matroid Theory, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text Matroid Theory, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

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    Barnard, A. D. Obituary: Aubrey William Ingleton (1920–2000). Bulletin of the London Mathematical Society, 36 (2004), 119129.

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    Brualdi, R. A. Common Transversals and Strong Exchange Systems. Journal of Combinatorial Theory, 8 (1970), 307329.

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    Oxley, J. The Contributions of Dominic Welsh to Matroid Theory. In Geo_rey Grimmett and Colin McDiarmid (eds.), Combinatorics, Complexity, and Chance. A Tribute to Dominic Welsh. Oxford University Press, Oxford, 2007, 234259.

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    Perfect, H. Remark on a Criterion for Common Transversals. Glasgow Mathematical Journal, 10 (1969), 6667.

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    Pym, J. S. and Perfect, H. Submodular Functions and Independence Structures. Journal of Mathematical Analysis and Applications, 30 (1970), 131.

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    Steffens, K. Ergebnisse aus der Transversalentheorie. II. Journal of Combinatorial Theory (A), 20 (1976), 202229.

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    Welsh, D. J. A . Matroid Theory. Academic Press, Inc., New York City, 1976.

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Mathematica Pannonica
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