Skip to main content
Top

2013 | OriginalPaper | Chapter

On Some Hypergraph Problems of Paul Erdős and the Asymptotics of Matchings, Covers and Colorings

Author : Jeff Kahn

Published in: The Mathematics of Paul Erdős I

Publisher: Springer New York

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This article summarizes progress on several old hypergraph problems of Paul Erdős and a few questions to which they led. Quite unexpectedly, there turned out to be substantial connections between the problems under discussion, surely some indication (if any were needed) that Erdős’ questions were the “right” ones. Here’s a quick synopsis.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference M. Ajtai, J. Komlós and E. Szemerédi, A dense infinite Sidon sequence, Europ. J. Combinatorics 2 (1981), 1–11. M. Ajtai, J. Komlós and E. Szemerédi, A dense infinite Sidon sequence, Europ. J. Combinatorics 2 (1981), 1–11.
2.
go back to reference M. Ajtai, J. Komlós and E. Szemerédi, A note on Ramsey numbers, J. Combinatorial Th. (A) 299 (1980), 354–360. M. Ajtai, J. Komlós and E. Szemerédi, A note on Ramsey numbers, J. Combinatorial Th. (A) 299 (1980), 354–360.
3.
go back to reference N. Alon, Restricted colorings of graphs, Surveys in Combinatorics, 1993 (Proc. 14th British Combinatorial Conf.), Cambridge Univ. Press, Cambridge, 1993. N. Alon, Restricted colorings of graphs, Surveys in Combinatorics, 1993 (Proc. 14th British Combinatorial Conf.), Cambridge Univ. Press, Cambridge, 1993.
4.
go back to reference N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, New York, 1992.MATH N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, New York, 1992.MATH
5.
go back to reference N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), 125–134. N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), 125–134.
6.
go back to reference L.D. Andersen, On edge-colourings of graphs, Math. Scand. 40 (1977), 161–175. L.D. Andersen, On edge-colourings of graphs, Math. Scand. 40 (1977), 161–175.
7.
go back to reference K. Azuma, Weighted sums of certain dependent random variables, Tokuku Math. J. 19 (1967), 357–367. K. Azuma, Weighted sums of certain dependent random variables, Tokuku Math. J. 19 (1967), 357–367.
8.
go back to reference E. A. Bender and E. R. Canfield, The asymptotic number of labelled graphs with given degree sequences, J. Combinatorial Th. (A) 24 (1978), 296–307. E. A. Bender and E. R. Canfield, The asymptotic number of labelled graphs with given degree sequences, J. Combinatorial Th. (A) 24 (1978), 296–307.
9.
go back to reference C. Berge, Hypergraphs: Combinatorics of Finite Sets, North Holland, Amsterdam, 1989.MATH C. Berge, Hypergraphs: Combinatorics of Finite Sets, North Holland, Amsterdam, 1989.MATH
10.
go back to reference C. Berge, On the chromatic index of a linear hypergraph and the Chvátal conjecture, pp. 40–44 in Combinatorial Mathematics: Proc. 3rd Int’l. Conf (New York, 1985), Ann. N. Y. Acad. Sci. 555, New York Acad. Sci., New York, 1989. C. Berge, On the chromatic index of a linear hypergraph and the Chvátal conjecture, pp. 40–44 in Combinatorial Mathematics: Proc. 3rd Int’l. Conf (New York, 1985), Ann. N. Y. Acad. Sci. 555, New York Acad. Sci., New York, 1989.
11.
go back to reference A. Blokhuis, More on maximal intersecting families of finite sets, J. Combinatorial Th. (A) 44 (1987), 299–303. A. Blokhuis, More on maximal intersecting families of finite sets, J. Combinatorial Th. (A) 44 (1987), 299–303.
12.
go back to reference B. Bollobás, A probabilistic proof of an asymptotic formula for the number of regular graphs, Europ. J. Combinatorics 1 (1980), 311–316. B. Bollobás, A probabilistic proof of an asymptotic formula for the number of regular graphs, Europ. J. Combinatorics 1 (1980), 311–316.
13.
go back to reference B. Bollobás, The independence ratio of regular graphs, Proc. Amer. Math. Soc. 83 (1981), 433–436. B. Bollobás, The independence ratio of regular graphs, Proc. Amer. Math. Soc. 83 (1981), 433–436.
14.
go back to reference B. Bollobás, Martingales, isoperimetric inequalities and random graphs, in Combinatories, A. Hajnal, L. Lovász and V.T. Sós Eds., Colloq. Math. Soc. János Bolyai 52, North Holland, 1988. B. Bollobás, Martingales, isoperimetric inequalities and random graphs, in Combinatories, A. Hajnal, L. Lovász and V.T. Sós Eds., Colloq. Math. Soc. János Bolyai 52, North Holland, 1988.
15.
go back to reference B. Bollobás and A. J. Harris, List-colourings of graphs, Graphs and Combinatorics 1 (1985), 115–127. B. Bollobás and A. J. Harris, List-colourings of graphs, Graphs and Combinatorics 1 (1985), 115–127.
16.
go back to reference B. Bollobás and H. Hind, A new upper bound for the list chromatic number, Discrete Math. 74 (1989), 65–75. B. Bollobás and H. Hind, A new upper bound for the list chromatic number, Discrete Math. 74 (1989), 65–75.
17.
go back to reference V. Boltjansky and I. Gohberg, Results and Problems in Combinatorial Geometry, Cambridge University Press, Cambridge, 1985.MATHCrossRef V. Boltjansky and I. Gohberg, Results and Problems in Combinatorial Geometry, Cambridge University Press, Cambridge, 1985.MATHCrossRef
18.
go back to reference E. Boros, Z. Füredi and J. Kahn, Maximal intersecting families and affine regular polygons, J. Combinatorial Th. (A) 52 (1989), 1–9. E. Boros, Z. Füredi and J. Kahn, Maximal intersecting families and affine regular polygons, J. Combinatorial Th. (A) 52 (1989), 1–9.
19.
go back to reference K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fundamenta Math. 20 (1933), 177–190. K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fundamenta Math. 20 (1933), 177–190.
20.
go back to reference N. G. de Bruijn and P. Erdős, A combinatorial problem, Indagationes Math. 8 (1946), 461–467. N. G. de Bruijn and P. Erdős, A combinatorial problem, Indagationes Math. 8 (1946), 461–467.
21.
go back to reference W. I. Chang and E. Lawler, Edge coloring of hypergraphs and a conjecture of Erdős, Faber, Lovász, Combinatorica 8 (1989), 293–295. W. I. Chang and E. Lawler, Edge coloring of hypergraphs and a conjecture of Erdős, Faber, Lovász, Combinatorica 8 (1989), 293–295.
22.
go back to reference A. Chetwynd and R. Häggkvist, A note on list-colorings, J. Graph Th. 13 (1989), 87–95. A. Chetwynd and R. Häggkvist, A note on list-colorings, J. Graph Th. 13 (1989), 87–95.
23.
go back to reference S. Chowla, P. Erdős and E. Straus, On the maximal number of pairwise orthogonal Latin squares of a given order, Can. J. Math. 12 (1960), 204–208.MATHCrossRef S. Chowla, P. Erdős and E. Straus, On the maximal number of pairwise orthogonal Latin squares of a given order, Can. J. Math. 12 (1960), 204–208.MATHCrossRef
24.
go back to reference H. Croft, K. Falconer and R. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.MATHCrossRef H. Croft, K. Falconer and R. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.MATHCrossRef
25.
go back to reference S. J. Dow, C. A. Drake, Z. Füredi and J. A. Larson, A lower bound for the cardinality of a maximal family of mutually intersecting sets of equal size, Congressus Numerantium 48 (1985), 47–48. S. J. Dow, C. A. Drake, Z. Füredi and J. A. Larson, A lower bound for the cardinality of a maximal family of mutually intersecting sets of equal size, Congressus Numerantium 48 (1985), 47–48.
26.
go back to reference J. Edmonds, Maximum matching and a polyhedron with 0, 1-vertices, J. Res. Nat. Bureau of Standards (B) 69 (1965), 125–130. J. Edmonds, Maximum matching and a polyhedron with 0, 1-vertices, J. Res. Nat. Bureau of Standards (B) 69 (1965), 125–130.
27.
go back to reference M.N. Ellingham and L. Goddyn, List edge colourings of some regular 1-factorable multigraphs, Combinatorica16 (1996), 343–352. M.N. Ellingham and L. Goddyn, List edge colourings of some regular 1-factorable multigraphs, Combinatorica16 (1996), 343–352.
28.
go back to reference P. Erdős, Some old and new problems in various branches of combinatorics, Congressus Numerantium 23 (1979), 19–37. P. Erdős, Some old and new problems in various branches of combinatorics, Congressus Numerantium 23 (1979), 19–37.
29.
go back to reference P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42. P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42.
30.
go back to reference P. Erdős, My Scottish book “problems”, pp. 35–43 in The Scottish Book. Mathematics from the Scottish Café (R.D. Mauldin, ed.), Birkhäuser, 1981. P. Erdős, My Scottish book “problems”, pp. 35–43 in The Scottish Book. Mathematics from the Scottish Café (R.D. Mauldin, ed.), Birkhäuser, 1981.
31.
go back to reference P. Erdős, Some of my old and new combinatorial problems, pp. 35–46 in Paths, Flows and VLSI-Layout (B. Korte, L. Lovász, H.-J. Promel and A. Schrijver, eds.), Springer, Berlin, 1990. P. Erdős, Some of my old and new combinatorial problems, pp. 35–46 in Paths, Flows and VLSI-Layout (B. Korte, L. Lovász, H.-J. Promel and A. Schrijver, eds.), Springer, Berlin, 1990.
32.
go back to reference P. Erdős and H. Hanani, On a limit theorem in combinatorial analysis, Publ. Math. Debrecen 10 (1963), 10–13. P. Erdős and H. Hanani, On a limit theorem in combinatorial analysis, Publ. Math. Debrecen 10 (1963), 10–13.
33.
go back to reference P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, Coll. Math. Soc. J. Bolyai 10 (1974), 609–627. P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, Coll. Math. Soc. J. Bolyai 10 (1974), 609–627.
34.
go back to reference P. Erdős, A. Rubin and H. Taylor, Choosability in graphs, Congressus Numerantium 26 (1979), 125–157. P. Erdős, A. Rubin and H. Taylor, Choosability in graphs, Congressus Numerantium 26 (1979), 125–157.
35.
go back to reference S. J. Fiorini and R. J. Wilson, Edge Colourings of Graphs, Research Notes in Mathematics 16, Pitman, London, 1977. S. J. Fiorini and R. J. Wilson, Edge Colourings of Graphs, Research Notes in Mathematics 16, Pitman, London, 1977.
36.
go back to reference R. A. Fisher, An examination of the different possible solutions of a problem in incomplete blocks, Ann Eugenics 10 (1940), 52–75. R. A. Fisher, An examination of the different possible solutions of a problem in incomplete blocks, Ann Eugenics 10 (1940), 52–75.
37.
go back to reference H. Fleischner and M. Stiebitz, A solution to a colouring problem of P. Erdős, Disc. Math. 101 (1992), 39–48. H. Fleischner and M. Stiebitz, A solution to a colouring problem of P. Erdős, Disc. Math. 101 (1992), 39–48.
38.
go back to reference P. Frankl and V. Rödl, Near-perfect coverings in graphs and hypergraphs, Europ. J. Combinatorics 6 (1985), 317–326. P. Frankl and V. Rödl, Near-perfect coverings in graphs and hypergraphs, Europ. J. Combinatorics 6 (1985), 317–326.
39.
go back to reference P. Frankl and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368. P. Frankl and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368.
40.
go back to reference Z. Füredi, On maximal intersecting families of finite sets, J. Combinatorial Th. (A) 28 (1980), 282–289. Z. Füredi, On maximal intersecting families of finite sets, J. Combinatorial Th. (A) 28 (1980), 282–289.
41.
go back to reference Z. Füredi, The chromatic index of simple hypergraphs, Graphs and Combinetorics 2 (1986), 89–92. Z. Füredi, The chromatic index of simple hypergraphs, Graphs and Combinetorics 2 (1986), 89–92.
42.
go back to reference Z. Füredi, Matchings and covers in hypergraphs, Graphs and Combinatorics 4 (1988), 115–206. Z. Füredi, Matchings and covers in hypergraphs, Graphs and Combinatorics 4 (1988), 115–206.
43.
go back to reference C. D. Godsil, Matching behavior is asymptotically normal, Combinatorica 1 (1981), 369–376. C. D. Godsil, Matching behavior is asymptotically normal, Combinatorica 1 (1981), 369–376.
44.
go back to reference C.D. Godsil, Matchings and walks in graphs, J. Graph Th. 5 (1981), 285–297. C.D. Godsil, Matchings and walks in graphs, J. Graph Th. 5 (1981), 285–297.
45.
go back to reference M. K. Goldberg, Edge-colorings of multigraphs: recoloring technique, J. Graph Th. 8 (1984), 123–127. M. K. Goldberg, Edge-colorings of multigraphs: recoloring technique, J. Graph Th. 8 (1984), 123–127.
46.
go back to reference B. Grünbaum, Borsuk’s problem and related questions, Proc. Symp. Pure Math. 7 (Convexity), Amer. Math. Soc., 1963. B. Grünbaum, Borsuk’s problem and related questions, Proc. Symp. Pure Math. 7 (Convexity), Amer. Math. Soc., 1963.
47.
go back to reference R. Häggkvist and A. Chetwynd, Some upper bounds on the total and list chromatic numbers of multigraphs, J. Graph Th. 16 (1992), 503–516. R. Häggkvist and A. Chetwynd, Some upper bounds on the total and list chromatic numbers of multigraphs, J. Graph Th. 16 (1992), 503–516.
48.
go back to reference R. Häggkvist and J. C. M. Janssen, On the list-chromatic index of bipartite graphs, manuscript, 1993. R. Häggkvist and J. C. M. Janssen, On the list-chromatic index of bipartite graphs, manuscript, 1993.
49.
go back to reference L.H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Stat. 38 (1967), 410–414. L.H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Stat. 38 (1967), 410–414.
50.
go back to reference O. J. Heilmann and E.H. Lieb, Monomers and dimers, Phys. Rev. Letters 24 (1970), 1412–1414. O. J. Heilmann and E.H. Lieb, Monomers and dimers, Phys. Rev. Letters 24 (1970), 1412–1414.
51.
go back to reference O.J. Heilmann and E.H. Lieb, Theory of monomer-dimer systems, Comm. Math. Physics 25 (1972), 190–232. O.J. Heilmann and E.H. Lieb, Theory of monomer-dimer systems, Comm. Math. Physics 25 (1972), 190–232.
52.
go back to reference H.R. Hind, Restricted edge-colourings, Doctoral thesis, Peterhouse College, Cambridge, 1988. H.R. Hind, Restricted edge-colourings, Doctoral thesis, Peterhouse College, Cambridge, 1988.
53.
go back to reference N. Hindman, On a conjecture of Erdős, Faber and Lovász about n-colorings, Canadian J. Math. 33 (1981), 563–570. N. Hindman, On a conjecture of Erdős, Faber and Lovász about n-colorings, Canadian J. Math. 33 (1981), 563–570.
54.
go back to reference W. Hoeffding, Probability inequalities for sums of bounded random variables. J. Amer. Stat. Assoc. 27 (1963), 13–30. W. Hoeffding, Probability inequalities for sums of bounded random variables. J. Amer. Stat. Assoc. 27 (1963), 13–30.
55.
go back to reference R. Huang and G.-C. Rota, On the relations of various conjectures on Latin squares and straightening coefficients, manuscript, 1993. R. Huang and G.-C. Rota, On the relations of various conjectures on Latin squares and straightening coefficients, manuscript, 1993.
56.
go back to reference J.C.M. Janssen, The Dinitz problem solved for rectangles, Bull. Amer. Math. Soc. 29 (1993), 243–249. J.C.M. Janssen, The Dinitz problem solved for rectangles, Bull. Amer. Math. Soc. 29 (1993), 243–249.
57.
go back to reference J. Kahn, On a theorem of Frankl and Rödl, in preparation. J. Kahn, On a theorem of Frankl and Rödl, in preparation.
58.
go back to reference J. Kahn, Coloring nearly-disjoint hypergraphs with n + o(n) colors, J. Combinatorial Th. (A) 59 (1992), 31–39. J. Kahn, Coloring nearly-disjoint hypergraphs with n + o(n) colors, J. Combinatorial Th. (A) 59 (1992), 31–39.
59.
go back to reference J. Kahn, On a problem of Erdős and Lovász: random lines in a projective plane, Combinatorica 12 (1992), 417–423. J. Kahn, On a problem of Erdős and Lovász: random lines in a projective plane, Combinatorica 12 (1992), 417–423.
60.
go back to reference J. Kahn, Recent results on some not-so-recent hypergraph matching and covering problems, pp. 305–353 in Extremal Problems for Finite Sets, Visegrád, 1991, Bolyai Soc. Math. Studies 3, 1994. J. Kahn, Recent results on some not-so-recent hypergraph matching and covering problems, pp. 305–353 in Extremal Problems for Finite Sets, Visegrád, 1991, Bolyai Soc. Math. Studies 3, 1994.
61.
go back to reference J. Kahn, On a problem of Erdős and Lovász II: n(r) = O(r), J. Amer. Math. Soc. 7 (1994), 125–143. J. Kahn, On a problem of Erdős and Lovász II: n(r) = O(r), J. Amer. Math. Soc. 7 (1994), 125–143.
63.
go back to reference J. Kahn, A normal law for matchings, Combinatorica 20 (2000), 339–391. J. Kahn, A normal law for matchings, Combinatorica 20 (2000), 339–391.
64.
go back to reference J. Kahn and G. Kalai, A problem of Füredi and Seymour on covering intersecting families by pairs, J. Combinatorial Th. 68 (1994), 317–339.MathSciNetMATHCrossRef J. Kahn and G. Kalai, A problem of Füredi and Seymour on covering intersecting families by pairs, J. Combinatorial Th. 68 (1994), 317–339.MathSciNetMATHCrossRef
65.
go back to reference J. Kahn and G. Kalai, A counterexample to Borsuk’s Conjecture, Bull. Amer. Math. Soc. 29 (1993), 60–62. J. Kahn and G. Kalai, A counterexample to Borsuk’s Conjecture, Bull. Amer. Math. Soc. 29 (1993), 60–62.
66.
go back to reference J. Kahn and M. Kayll, Fractional vs. integral covers in hypergraphs of bounded edge size, J. Combinatorial Th. A 78 (1997), 199–235. J. Kahn and M. Kayll, Fractional vs. integral covers in hypergraphs of bounded edge size, J. Combinatorial Th. A 78 (1997), 199–235.
67.
go back to reference J. Kahn and J.H. Kim, Random matchings in regular graphs, Combinatorica 18 (1998), 201–226. J. Kahn and J.H. Kim, Random matchings in regular graphs, Combinatorica 18 (1998), 201–226.
68.
go back to reference J. Kahn and P. D. Seymour, A fractional version of the Erdős-Faber-Lovász Conjecture, Combinatorica 12 (1992), 155–160. J. Kahn and P. D. Seymour, A fractional version of the Erdős-Faber-Lovász Conjecture, Combinatorica 12 (1992), 155–160.
69.
go back to reference J. H. Kim, On Brooks’ Theorem for sparse graphs, Combinatorics, Probability and Computing 4 (1995), 97–132. J. H. Kim, On Brooks’ Theorem for sparse graphs, Combinatorics, Probability and Computing 4 (1995), 97–132.
70.
go back to reference J. Komlós, J. Pintz and E. Szemerédi, A lower bound for Heilbronn’s problem, J. London Math. Soc. 25 (1982), 13–24. J. Komlós, J. Pintz and E. Szemerédi, A lower bound for Heilbronn’s problem, J. London Math. Soc. 25 (1982), 13–24.
71.
go back to reference A. V. Kostochka, Degree, girth and chromatic number, pp. 679–696 in Combinatorics, Keszthely 1976, A. Hajnal and V. T. Sós Eds., Colloq. Math. Soc. János Bolyai 18, North Holland, 1978. A. V. Kostochka, Degree, girth and chromatic number, pp. 679–696 in Combinatorics, Keszthely 1976, A. Hajnal and V. T. Sós Eds., Colloq. Math. Soc. János Bolyai 18, North Holland, 1978.
72.
go back to reference H. Kunz, Location of the zeros of the partition function for some classical lattice systems, Phys. Lett. (A) (1970), 311–312. H. Kunz, Location of the zeros of the partition function for some classical lattice systems, Phys. Lett. (A) (1970), 311–312.
73.
go back to reference D. G. Larman, in Convexity and Graph Theory (Rosenfeld and Zaks, eds.), Ann. Discrete Math. 20 (1984), 336. D. G. Larman, in Convexity and Graph Theory (Rosenfeld and Zaks, eds.), Ann. Discrete Math. 20 (1984), 336.
74.
go back to reference L. Lovász and M.D. Plummer, Matching Theory, North Holland, Amsterdam, 1986.MATH L. Lovász and M.D. Plummer, Matching Theory, North Holland, Amsterdam, 1986.MATH
75.
go back to reference C.J.H. McDiarmid, On the method of bounded differences, pp. 148–188 in Surveys in Combinatorics 1989, Invited Papers at the 12th British Combinatorial Conference, J. Siemons Ed., Cambridge Univ. Pr., 1989. C.J.H. McDiarmid, On the method of bounded differences, pp. 148–188 in Surveys in Combinatorics 1989, Invited Papers at the 12th British Combinatorial Conference, J. Siemons Ed., Cambridge Univ. Pr., 1989.
76.
go back to reference J.-C. Meyer, pp. 285–286 in Hypergraph Seminar (Berge and Ray-Chaudhuri, eds.), Springer, Berlin-Heidelberg-New York, 1974. J.-C. Meyer, pp. 285–286 in Hypergraph Seminar (Berge and Ray-Chaudhuri, eds.), Springer, Berlin-Heidelberg-New York, 1974.
77.
go back to reference N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combinatorial Th. (A) 51 (1989), 24–42. N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combinatorial Th. (A) 51 (1989), 24–42.
78.
go back to reference Y. Rabinovich, A. Sinclair and A. Wigderson, Quadratic Dynamical Systems, Proc. 33rd IEEE Symposium on Foundations of Computer Science (1992), 304–313. Y. Rabinovich, A. Sinclair and A. Wigderson, Quadratic Dynamical Systems, Proc. 33rd IEEE Symposium on Foundations of Computer Science (1992), 304–313.
79.
go back to reference J. Riordan, An Introduction to Combinatorial Analysis, Wiley, New York, 1958.MATH J. Riordan, An Introduction to Combinatorial Analysis, Wiley, New York, 1958.MATH
80.
go back to reference V. Rödl, On a packing and covering problem, Europ. J. Combinatorics 5 (1985), 69–78. V. Rödl, On a packing and covering problem, Europ. J. Combinatorics 5 (1985), 69–78.
81.
go back to reference A. Ruciński, The behaviour of \(\left ({ n \atop k,,\ldots,,k,n-ik} \right ){c}^{i}/i!\) is asymptotically normal, Discrete Math. 49 (1984), 287–290. A. Ruciński, The behaviour of \(\left ({ n \atop k,,\ldots,,k,n-ik} \right ){c}^{i}/i!\) is asymptotically normal, Discrete Math. 49 (1984), 287–290.
82.
go back to reference A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986.MATH A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986.MATH
83.
go back to reference P. D. Seymour, Some unsolved problems on one-factorizations of graphs, in Graph Theory and Related Topics (Bondy and Murty, eds.), Academic Press, New York, 1979. P. D. Seymour, Some unsolved problems on one-factorizations of graphs, in Graph Theory and Related Topics (Bondy and Murty, eds.), Academic Press, New York, 1979.
84.
go back to reference P. D. Seymour, Packing nearly-disjoint sets, Combinatorica 2 (1982), 91–97. P. D. Seymour, Packing nearly-disjoint sets, Combinatorica 2 (1982), 91–97.
85.
go back to reference C. E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949), 148–151. C. E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949), 148–151.
87.
88.
89.
go back to reference B. Toft, 75 graph-colouring problems, pp. 9–35 in Graph Colourings (R. Nelson and R.J. Wilson, eds.), Wiley, New York, 1990. B. Toft, 75 graph-colouring problems, pp. 9–35 in Graph Colourings (R. Nelson and R.J. Wilson, eds.), Wiley, New York, 1990.
90.
go back to reference V. G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964), 25–30. V. G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964), 25–30.
91.
go back to reference V. G. Vizing, Some unsolved problems in graph theory (Russian), Uspehi Mat. Nauk. 23 (1968), 117–134. English translation in Russian Math. Surveys 23 (1968), 125–141. V. G. Vizing, Some unsolved problems in graph theory (Russian), Uspehi Mat. Nauk. 23 (1968), 117–134. English translation in Russian Math. Surveys 23 (1968), 125–141.
92.
go back to reference V. G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz No 29 Metody Diskret. Anal. v Teorii Kodov i Shem (1976), 3–10, 101 (MR58 #16371). V. G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz No 29 Metody Diskret. Anal. v Teorii Kodov i Shem (1976), 3–10, 101 (MR58 #16371).
93.
go back to reference R. M. Wilson, An existence theory for pairwise balanced designs I–III, J. Combinatorial Th. (A) 13 (1972), 220–273, 18 (1975), 71–79. R. M. Wilson, An existence theory for pairwise balanced designs I–III, J. Combinatorial Th. (A) 13 (1972), 220–273, 18 (1975), 71–79.
Metadata
Title
On Some Hypergraph Problems of Paul Erdős and the Asymptotics of Matchings, Covers and Colorings
Author
Jeff Kahn
Copyright Year
2013
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-7258-2_22

Premium Partner