[Journal] The Mathematical Intelligencer. Vol. 32. No 2

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J. Wilson), Cambridge University Press (2009), 133–150. [25] Paul Tura´n, A note of welcome, J. Graph Theory 1 (1977), 7–9. [26] K. Urbanik, Solution du proble`me pose´ par P. Tura´n, Colloq. [15] Anthony Hill, Catalogue of the Retrospective Exhibition, Arts Council of Great Britain, Hayward Gallery, 1983. [16] D. J. Kleitman, The crossing number of K5,n, J. Combin. Theory 9 (1970), 315–323. [17] T. Ko¨vari, V. So´s, and P. Tura´n: On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50–57.

Finding the crossing numbers of the complete graphs has a more confused history than that of the complete bipartite graphs. For one thing, the complete graph problem seems to be a more natural place to start and various people may have considered it until its difficulty discouraged them from pursuing it. Those who heard Tura´n describe the brick factory problem may also have thought about this problem; certainly Paul Erd} os claimed in 1960 to have been looking at the problem for at least 20 years, but uncharacteristically seems to have told no one else about it.

E. J. Brouwer in April of that year. Brouwer was of the opinion that the crossingnumber problem might be like the four-colour problem and present great difficulties, in spite of its simple sounding nature. In May 1959, Hill communicated the problem to Professor Ambrose Rogers of University College, the geometer John Todd in Cambridge, and the combinatorialist Richard Rado at the University of Reading. At Rado’s suggestion, he wrote also to the French graph-theorist Claude Berge. Rado believed the problem to be difficult, but no one could shed any light on it.

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