De Bruijn–Erdős theorem (graph theory) (English Wikipedia)
Analysis of information sources in references of the Wikipedia article "De Bruijn–Erdős theorem (graph theory)" in English language version.
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- For these basic definitions, see Jensen & Toft (1995), pp. 1–2. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254.
- Jensen & Toft (1995), p. 5. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254.
- Jensen & Toft (1995), Theorem 1, p. 2. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254.
- Erdős (1950). See in particular p. 137, where the De Bruijn–Erdős theorem is first announced (but not proven), with a hint that Kőnig's lemma can be used for countable graphs. Erdős, P. (1950), "Some remarks on set theory" (PDF), Proceedings of the American Mathematical Society, 1 (2): 127–141, doi:10.2307/2031913, JSTOR 2031913, MR 0035809.
- Harzheim (2005), Theorem 5.6, p. 60. Harzheim, Egbert (2005), Ordered sets, Advances in Mathematics, vol. 7, New York: Springer, Theorem 5.5, p. 59, ISBN 0-387-24219-8, MR 2127991.
- Barnette (1983). Nash-Williams (1967) states the same result for the five-color theorem for countable planar graphs, as the four-color theorem had not yet been proven when he published his survey, and as the proof of the De Bruijn–Erdős theorem that he gives only applies to countable graphs. For the generalization to graphs in which every finite subgraph is planar (proved directly via Gödel's compactness theorem), see Rautenberg (2010). Nash-Williams, C. St. J. A. (1967), "Infinite graphs—a survey", Journal of Combinatorial Theory, 3 (3): 286–301, doi:10.1016/s0021-9800(67)80077-2, MR 0214501. Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic, Universitext (3rd ed.), Springer-Verlag, p. 32, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.
- Jensen & Toft (1995). Gottschalk states his proof more generally as a proof of the theorem of Rado (1949) that generalizes the De Bruijn–Erdős theorem. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254. Rado, R. (1949), "Axiomatic treatment of rank in infinite sets", Canadian Journal of Mathematics, 1 (4): 337–343, doi:10.4153/cjm-1949-031-1, S2CID 124598982.
- Jensen & Toft (1995); Harzheim (2005). Jensen and Toft attribute to Gert Sabidussi the observation that Gottschalk's proof is easier to generalize. Soifer (2008, pp. 238–239) gives the same proof via Zorn's lemma, rediscovered in 2005 by undergraduate student Dmytro Karabash. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254. Harzheim, Egbert (2005), Ordered sets, Advances in Mathematics, vol. 7, New York: Springer, Theorem 5.5, p. 59, ISBN 0-387-24219-8, MR 2127991. Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1. See especially Chapter II.5 "De Bruin–Erdős reduction to finite sets and results near the lower bound", pp. 39–42, and Chapter V.26 "De Bruin–Erdős's theorem and its history", pp. 236–241.
- Luxemburg (1962). Luxemburg, W. A. J. (1962), "A remark on a paper by N. G. de Bruijn and P. Erdős", Indagationes Mathematicae, 24: 343–345, doi:10.1016/S1385-7258(62)50033-4, MR 0140435.
- Hurd & Loeb (1985). Hurd, Albert E.; Loeb, Peter A. (1985), An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Orlando, FL: Academic Press, Theorem 5.14, p. 92, ISBN 0-12-362440-1, MR 0806135.
- For this history, see Cowen, Hechler & Mihók (2002). For a simplified proof of Läuchli's theorem by Mycielski, see Cowen (1990). Cowen, Robert; Hechler, Stephen H.; Mihók, Peter (2002), "Graph coloring compactness theorems equivalent to BPI", Scientiae Mathematicae Japonicae, 56 (2): 213–223, CiteSeerX 10.1.1.23.1521, MR 1922784. Cowen, Robert H. (1990), "Two hypergraph theorems equivalent to BPI", Notre Dame Journal of Formal Logic, 31 (2): 232–240, doi:10.1305/ndjfl/1093635418, MR 1058424.
- Schmerl (2000). Schmerl, James H. (2000), "Graph coloring and reverse mathematics", Mathematical Logic Quarterly, 46 (4): 543–548, doi:10.1002/1521-3870(200010)46:4<543::AID-MALQ543>3.0.CO;2-E, MR 1791549.
- Shelah & Soifer (2003); Soifer (2008), pp. 541–542. Shelah, Saharon; Soifer, Alexander (2003), "Axiom of choice and chromatic number of the plane", Journal of Combinatorial Theory, Series A, 103 (2): 387–391, doi:10.1016/S0097-3165(03)00102-X, MR 1996076. Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1. See especially Chapter II.5 "De Bruin–Erdős reduction to finite sets and results near the lower bound", pp. 39–42, and Chapter V.26 "De Bruin–Erdős's theorem and its history", pp. 236–241.
- For this connection between Rado's lemma and the De Bruijn–Erdős theorem, see e.g. the discussion following Theorem A of Nash-Williams (1967). Nash-Williams, C. St. J. A. (1967), "Infinite graphs—a survey", Journal of Combinatorial Theory, 3 (3): 286–301, doi:10.1016/s0021-9800(67)80077-2, MR 0214501.
- Erdős & Hajnal (1966); Komjáth (2011). Erdős, P.; Hajnal, A. (1966), "On chromatic number of graphs and set-systems" (PDF), Acta Mathematica Academiae Scientiarum Hungaricae, 17 (1–2): 61–99, doi:10.1007/BF02020444, MR 0193025, S2CID 189780485. Komjáth, Péter (2011), "The chromatic number of infinite graphs—A survey", Discrete Mathematics, 311 (15): 1448–1450, doi:10.1016/j.disc.2010.11.004.
- Lake (1975), p. 171: "It is straightforward to prove [the De Bruijn–Erdős theorem] using the compactness theorem for first-order logic" Lake, John (1975), "A generalization of a theorem of de Bruijn and Erdős on the chromatic number of infinite graphs", Journal of Combinatorial Theory, Series B, 18 (2): 170–174, doi:10.1016/0095-8956(75)90044-1, MR 0360335.
- Rorabaugh, Tardif & Wehlau (2017). Rorabaugh, Danny; Tardif, Claude; Wehlau, David (2017), "Logical compactness and constraint satisfaction problems", Logical Methods in Computer Science, 13 (1): 1:1–1:11, arXiv:1609.05221, doi:10.23638/LMCS-13(1:1)2017, MR 3607036, S2CID 31135533.
- Erdős & Hajnal (1968). Erdős, P.; Hajnal, A. (1968), "On chromatic number of infinite graphs", Theory of Graphs (Proc. Colloq., Tihany, 1966) (PDF), New York: Academic Press, pp. 83–98, MR 0263693.
arxiv.org
- Rorabaugh, Tardif & Wehlau (2017). Rorabaugh, Danny; Tardif, Claude; Wehlau, David (2017), "Logical compactness and constraint satisfaction problems", Logical Methods in Computer Science, 13 (1): 1:1–1:11, arXiv:1609.05221, doi:10.23638/LMCS-13(1:1)2017, MR 3607036, S2CID 31135533.
books.google.com
- Harzheim (2005), Theorem 5.6, p. 60. Harzheim, Egbert (2005), Ordered sets, Advances in Mathematics, vol. 7, New York: Springer, Theorem 5.5, p. 59, ISBN 0-387-24219-8, MR 2127991.
- Jensen & Toft (1995); Harzheim (2005). Jensen and Toft attribute to Gert Sabidussi the observation that Gottschalk's proof is easier to generalize. Soifer (2008, pp. 238–239) gives the same proof via Zorn's lemma, rediscovered in 2005 by undergraduate student Dmytro Karabash. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254. Harzheim, Egbert (2005), Ordered sets, Advances in Mathematics, vol. 7, New York: Springer, Theorem 5.5, p. 59, ISBN 0-387-24219-8, MR 2127991. Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1. See especially Chapter II.5 "De Bruin–Erdős reduction to finite sets and results near the lower bound", pp. 39–42, and Chapter V.26 "De Bruin–Erdős's theorem and its history", pp. 236–241.
- Hurd & Loeb (1985). Hurd, Albert E.; Loeb, Peter A. (1985), An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Orlando, FL: Academic Press, Theorem 5.14, p. 92, ISBN 0-12-362440-1, MR 0806135.
- This follows immediately from the definition of a strongly compact cardinal as being a cardinal such that every collection of formulae of infinitary logic each of length smaller than , that is satisfiable for every subcollection of fewer than formulae, is globally satisfiable. See e.g. Kanamori (2008). Kanamori, Akihiro (2008), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, p. 37, ISBN 978-3-540-88866-6.
doi.org
- Komjáth (2011). Komjáth, Péter (2011), "The chromatic number of infinite graphs—A survey", Discrete Mathematics, 311 (15): 1448–1450, doi:10.1016/j.disc.2010.11.004.
- Erdős (1950). See in particular p. 137, where the De Bruijn–Erdős theorem is first announced (but not proven), with a hint that Kőnig's lemma can be used for countable graphs. Erdős, P. (1950), "Some remarks on set theory" (PDF), Proceedings of the American Mathematical Society, 1 (2): 127–141, doi:10.2307/2031913, JSTOR 2031913, MR 0035809.
- Barnette (1983). Nash-Williams (1967) states the same result for the five-color theorem for countable planar graphs, as the four-color theorem had not yet been proven when he published his survey, and as the proof of the De Bruijn–Erdős theorem that he gives only applies to countable graphs. For the generalization to graphs in which every finite subgraph is planar (proved directly via Gödel's compactness theorem), see Rautenberg (2010). Nash-Williams, C. St. J. A. (1967), "Infinite graphs—a survey", Journal of Combinatorial Theory, 3 (3): 286–301, doi:10.1016/s0021-9800(67)80077-2, MR 0214501. Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic, Universitext (3rd ed.), Springer-Verlag, p. 32, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.
- Jensen & Toft (1995). Gottschalk states his proof more generally as a proof of the theorem of Rado (1949) that generalizes the De Bruijn–Erdős theorem. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254. Rado, R. (1949), "Axiomatic treatment of rank in infinite sets", Canadian Journal of Mathematics, 1 (4): 337–343, doi:10.4153/cjm-1949-031-1, S2CID 124598982.
- Luxemburg (1962). Luxemburg, W. A. J. (1962), "A remark on a paper by N. G. de Bruijn and P. Erdős", Indagationes Mathematicae, 24: 343–345, doi:10.1016/S1385-7258(62)50033-4, MR 0140435.
- For this history, see Cowen, Hechler & Mihók (2002). For a simplified proof of Läuchli's theorem by Mycielski, see Cowen (1990). Cowen, Robert; Hechler, Stephen H.; Mihók, Peter (2002), "Graph coloring compactness theorems equivalent to BPI", Scientiae Mathematicae Japonicae, 56 (2): 213–223, CiteSeerX 10.1.1.23.1521, MR 1922784. Cowen, Robert H. (1990), "Two hypergraph theorems equivalent to BPI", Notre Dame Journal of Formal Logic, 31 (2): 232–240, doi:10.1305/ndjfl/1093635418, MR 1058424.
- Schmerl (2000). Schmerl, James H. (2000), "Graph coloring and reverse mathematics", Mathematical Logic Quarterly, 46 (4): 543–548, doi:10.1002/1521-3870(200010)46:4<543::AID-MALQ543>3.0.CO;2-E, MR 1791549.
- Shelah & Soifer (2003); Soifer (2008), pp. 541–542. Shelah, Saharon; Soifer, Alexander (2003), "Axiom of choice and chromatic number of the plane", Journal of Combinatorial Theory, Series A, 103 (2): 387–391, doi:10.1016/S0097-3165(03)00102-X, MR 1996076. Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1. See especially Chapter II.5 "De Bruin–Erdős reduction to finite sets and results near the lower bound", pp. 39–42, and Chapter V.26 "De Bruin–Erdős's theorem and its history", pp. 236–241.
- For this connection between Rado's lemma and the De Bruijn–Erdős theorem, see e.g. the discussion following Theorem A of Nash-Williams (1967). Nash-Williams, C. St. J. A. (1967), "Infinite graphs—a survey", Journal of Combinatorial Theory, 3 (3): 286–301, doi:10.1016/s0021-9800(67)80077-2, MR 0214501.
- Erdős & Hajnal (1966); Komjáth (2011). Erdős, P.; Hajnal, A. (1966), "On chromatic number of graphs and set-systems" (PDF), Acta Mathematica Academiae Scientiarum Hungaricae, 17 (1–2): 61–99, doi:10.1007/BF02020444, MR 0193025, S2CID 189780485. Komjáth, Péter (2011), "The chromatic number of infinite graphs—A survey", Discrete Mathematics, 311 (15): 1448–1450, doi:10.1016/j.disc.2010.11.004.
- Lake (1975), p. 171: "It is straightforward to prove [the De Bruijn–Erdős theorem] using the compactness theorem for first-order logic" Lake, John (1975), "A generalization of a theorem of de Bruijn and Erdős on the chromatic number of infinite graphs", Journal of Combinatorial Theory, Series B, 18 (2): 170–174, doi:10.1016/0095-8956(75)90044-1, MR 0360335.
- Rorabaugh, Tardif & Wehlau (2017). Rorabaugh, Danny; Tardif, Claude; Wehlau, David (2017), "Logical compactness and constraint satisfaction problems", Logical Methods in Computer Science, 13 (1): 1:1–1:11, arXiv:1609.05221, doi:10.23638/LMCS-13(1:1)2017, MR 3607036, S2CID 31135533.
jstor.org
- Erdős (1950). See in particular p. 137, where the De Bruijn–Erdős theorem is first announced (but not proven), with a hint that Kőnig's lemma can be used for countable graphs. Erdős, P. (1950), "Some remarks on set theory" (PDF), Proceedings of the American Mathematical Society, 1 (2): 127–141, doi:10.2307/2031913, JSTOR 2031913, MR 0035809.
psu.edu
citeseerx.ist.psu.edu
- For this history, see Cowen, Hechler & Mihók (2002). For a simplified proof of Läuchli's theorem by Mycielski, see Cowen (1990). Cowen, Robert; Hechler, Stephen H.; Mihók, Peter (2002), "Graph coloring compactness theorems equivalent to BPI", Scientiae Mathematicae Japonicae, 56 (2): 213–223, CiteSeerX 10.1.1.23.1521, MR 1922784. Cowen, Robert H. (1990), "Two hypergraph theorems equivalent to BPI", Notre Dame Journal of Formal Logic, 31 (2): 232–240, doi:10.1305/ndjfl/1093635418, MR 1058424.
quantamagazine.org
- Lamb (2018). Lamb, Evelyn (April 17, 2018), "Decades-old graph problem yields to amateur mathematician", Quanta Magazine
renyi.hu
renyi.hu
- Erdős & Hajnal (1966); Komjáth (2011). Erdős, P.; Hajnal, A. (1966), "On chromatic number of graphs and set-systems" (PDF), Acta Mathematica Academiae Scientiarum Hungaricae, 17 (1–2): 61–99, doi:10.1007/BF02020444, MR 0193025, S2CID 189780485. Komjáth, Péter (2011), "The chromatic number of infinite graphs—A survey", Discrete Mathematics, 311 (15): 1448–1450, doi:10.1016/j.disc.2010.11.004.
- Erdős & Hajnal (1968). Erdős, P.; Hajnal, A. (1968), "On chromatic number of infinite graphs", Theory of Graphs (Proc. Colloq., Tihany, 1966) (PDF), New York: Academic Press, pp. 83–98, MR 0263693.
users.renyi.hu
- Erdős (1950). See in particular p. 137, where the De Bruijn–Erdős theorem is first announced (but not proven), with a hint that Kőnig's lemma can be used for countable graphs. Erdős, P. (1950), "Some remarks on set theory" (PDF), Proceedings of the American Mathematical Society, 1 (2): 127–141, doi:10.2307/2031913, JSTOR 2031913, MR 0035809.
semanticscholar.org
api.semanticscholar.org
- Jensen & Toft (1995). Gottschalk states his proof more generally as a proof of the theorem of Rado (1949) that generalizes the De Bruijn–Erdős theorem. Jensen, Tommy R.; Toft, Bjarne (1995), Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., Theorem 1, pp. 2–3, ISBN 0-471-02865-7, MR 1304254. Rado, R. (1949), "Axiomatic treatment of rank in infinite sets", Canadian Journal of Mathematics, 1 (4): 337–343, doi:10.4153/cjm-1949-031-1, S2CID 124598982.
- Erdős & Hajnal (1966); Komjáth (2011). Erdős, P.; Hajnal, A. (1966), "On chromatic number of graphs and set-systems" (PDF), Acta Mathematica Academiae Scientiarum Hungaricae, 17 (1–2): 61–99, doi:10.1007/BF02020444, MR 0193025, S2CID 189780485. Komjáth, Péter (2011), "The chromatic number of infinite graphs—A survey", Discrete Mathematics, 311 (15): 1448–1450, doi:10.1016/j.disc.2010.11.004.
- Rorabaugh, Tardif & Wehlau (2017). Rorabaugh, Danny; Tardif, Claude; Wehlau, David (2017), "Logical compactness and constraint satisfaction problems", Logical Methods in Computer Science, 13 (1): 1:1–1:11, arXiv:1609.05221, doi:10.23638/LMCS-13(1:1)2017, MR 3607036, S2CID 31135533.