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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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, 234–259.
Perfect, H. Remark on a Criterion for Common Transversals. Glasgow Mathematical Journal, 10 (1969), 66–67.
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Steffens, K. Ergebnisse aus der Transversalentheorie. II. Journal of Combinatorial Theory (A), 20 (1976), 202–229.
University of Waterloo, Citation for Dominic Welsh (2006). https://uwaterloo.ca/combinatoricsand-optimization/sites/ca.combinatorics-and-optimization/_les/uploads/_les/dom-award-1.pdf
Welsh, D. J. A. Matroid Theory. Academic Press, Inc., New York City, 1976.