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Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts

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Book cover Ramsey Theory

Part of the book series: Progress in Mathematics ((PM,volume 285))

Abstract

In August 1987 I attended an inspiring talk by Paul Halmos at Chapman College in Orange, California. It was entitled “Some Problems You Can Solve, and Some You Cannot.” This problem was an example of a problem “you cannot solve.”

Much but not all of this text is contained in the author’s monograph [Soi].

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Notes

  1. 1.

    [Bru6].

  2. 2.

    This seems to be my first mention of what has become an 18-year long project!

  3. 3.

    First publication could be attributed to De Morgan, who mentioned the problem in his 1860 book review in Athenaeum [DeM4], albeit anonymously.

  4. 4.

    Thanks to Prof. Fred Hoffman, the tireless organizer of this annual conference, I have a videotape of Paul Erdős’s memorable talk.

  5. 5.

    My translation from the Russian.

  6. 6.

    Ibid.

  7. 7.

    The Young Men’s Christian Association (YMCA) is one of the oldest and largest not-for-profit community service organizations in the world.

  8. 8.

    Robert Maynard Hutchins (1899–1977) was President (1929–1945) and Chancellor (1945–1951) of the University of Chicago.

  9. 9.

    Graham cites Paul O’Donnell’s Theorem 48.4 (see it later in this book) as “perhaps, the evidence that χ is at least 5.”

  10. 10.

    If the chromatic number of the plane is 7, then for G(x 1,…,xn)=7 such an n must be greater than 6197 [Pri].

  11. 11.

    The authors of the fine problem book [BMP] incorrectly credit Hadwiger as “first” to study this problem (p. 235). Hadwiger, quite typically for him, limited his study to partitions into closed sets.

  12. 12.

    Students in such high schools hold regular jobs during the day, and attend classes at night.

  13. 13.

    The axiom of choice is assumed in this result.

  14. 14.

    Or so we all thought until recently. Because of that, I chose to leave this section as it was written in the early 1990s. BUT: see Section X of this survey for the latest developments.

  15. 15.

    The symbol G(x 1,…,x n ) denotes the graph on the listed inside parentheses n vertices, with two vertices adjacent if and only if they are unit distance apart.

  16. 16.

    Symbol [a,b], a < b, as usual, stands for the line segment, including its endpoints a and b.

  17. 17.

    The important problem book [BMP] mistakenly cites only one of this series of three papers. It also incorrectly states that the authors proved only the lower bound 5, whereas they raised the lower bound to 6.

  18. 18.

    Curiously, Paul wrote an improbable date on the letter: “1977 VII 25”.

  19. 19.

    The De Bruijn–Erdős theorem assumes the axiom of choice.

  20. 20.

    Quoted from [Pet], p. 494.

  21. 21.

    It is the first task, but we did not think of it then, and so this definition appears for the first time in [Soi].

  22. 22.

    Quoted from [Pet], p. 494.

  23. 23.

    A cardinal κ is called inaccessible if κ>ℵ 0, κ is regular, and κ is strong limit. An infinite cardinal ℵ α is regular, if cfωα=ωα. A cardinal κ is a strong limit cardinal if for every cardinal λ, λ<κ implies2λ<κ.

  24. 24.

    Assuming the existence of an inaccessible cardinal.

  25. 25.

    Due to the use of the Solovay’s theorem, we assume the existence of an inaccessible cardinal.

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Soifer, A. (2011). Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts . In: Soifer, A. (eds) Ramsey Theory. Progress in Mathematics, vol 285. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8092-3_8

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