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Infinite Sets and Finite Combinatorics

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Abstract

In tackling the set-satisfiability problem in Chap. 4, we have not gone beyond the analysis of formulae with a single prefixed universal quantifier: we have seen how to determine whether or not a formula of the form \(\forall \,y\:\mu\) is satisfiable, where μ stands for a propositional combination of membership and equality literals.What happens if we allow multiple universal quantifiers to appear in the prefix of the formula subject to the set-satisfiability analysis? A legal input, now, is any formula \(\forall \,y_{1}\cdots \forall \,y_{m}\:\mu\), with μ devoid of quantifiers; to state it briefly, it is a ∀ ∗-formula which we want to make true via a substitution \(x\mapsto \boldsymbol{x}\) of sets to its free variables. We will devote this chapter to discussing the infinitudes which enter into the modeling of such \(\forall ^{{\ast}}\)-formulae.Our main motivation for undertaking this discussion is that the study of those infinitudes will provide a revealing insight on Ramsey’s celebrated theorem, fundamental in finite combinatorics. This also provides a good reason for not restraining our discussion to well-founded sets: by leaving out of the game the well-foundedness of membership, we will find a straighter bridge between the satisfiability problem for pure first-order logic and the set-satisfiability problem.

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Notes

  1. 1.

    We will feel free to ascribe to the BSR-class also equisatisfiable versions of Ψ 1 such as

    $$\displaystyle{\varPsi _{1}^{{\prime}}(x_{ 1},x_{2},x_{3}) \equiv \, x_{1} \in x_{3} \wedge x_{2}\notin x_{3} \wedge \forall y(\,y \in x_{1} \leftrightarrow y \in x_{2}\,),}$$

    or as

    $$\displaystyle{\varPsi _{1}^{{\prime\prime}}\equiv \exists x_{ 1},x_{2},x_{3}\forall \,y\big(\,x_{1} \in x_{3} \wedge x_{2}\notin x_{3} \wedge (\,y \in x_{1} \leftrightarrow y \in x_{2}\,)\,\big).}$$
  2. 2.

    We are making use of the union-set operator to increase readability: \(\bigcup a \subseteq b\) in fact abbreviates the BSR-formula \((\forall x \in a)(\forall y \in x)(y \in b)\).

  3. 3.

    We are focusing on ZF −− FA and Inf only momentarily and for the sake of definiteness: our considerations easily carry over to other axiomatic theories and to different statements of infinity.

  4. 4.

    We use the shorthand ⊆ in the context a ∈ b → b ⊆ a with the meaning \((\forall x)(a \in b \rightarrow (x \in b \rightarrow x \in a))\).

References

  1. Ackermann, W.: Die Widerspruchfreiheit der allgemeinen Mengenlehre. Mathematische Annalen 114, 305–315 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. Center for the Study of Language and Information, cop, Stanford (1988)

    Google Scholar 

  3. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1976)

    MATH  Google Scholar 

  4. Andersson, S.A., Madigan, D., Perlman, M.D.: A characterization of Markov equivalence classes for acyclic digraphs. Ann. Stat. 25, 502–541 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Audrito, G., Tomescu, A.I., Wagner, S.G.: Enumeration of the adjunctive hierarchy of hereditarily finite sets. J. Log. Comput. 25 (3), 943–963 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41 (2), 182–208 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer Monographs in Mathematics, 2nd edn. Springer, London (2008)

    Google Scholar 

  8. Barwise, J., Moss, L.: Hypersets. Math. Intell. 13 (4), 31–41 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Barwise, J., Moss, L.S.: Vicious Circles. CSLI Lecture Notes, Stanford (1996)

    MATH  Google Scholar 

  10. Behzad, M., Chartrand, G., Lesniak-Foster, L.: Graphs & Digraphs. Prindle, Weber & Schmidt (1979)

    MATH  Google Scholar 

  11. Belinfante, J.G.F.: On computer-assisted proofs in ordinal number theory. J. Autom. Reason. 22 (2), 341–378 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bellè, D., Parlamento, F.: Truth in V for ∃ ∗ ∀ ∀-sentences is decidable. J. Symb. Log. 71 (4), 1200–1222 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boyer, R.S., Lusk, E.L., McCune, W., Overbeek, R.A., Stickel, M.E., Wos, L.: Set theory in first-order logic: clauses for Gödel’s axioms. J. Autom. Reason. 2 (3), 287–327 (1986)

    Article  MATH  Google Scholar 

  14. Brandstädt, A., Klembt, T., Lozin, V.V., Mosca, R.: On independent vertex sets in subclasses of apple-free graphs. Algorithmica 56, 383–393 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Monographs on Discrete Mathematics and Applications, vol. 3. SIAM Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Google Scholar 

  16. Brandstädt, A., Lozin, V.V., Mosca, R.: Independent sets of maximum weight in apple-free graphs. SIAM J. Discret. Math. 24, 239–254 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Burstall, R., Goguen, J.: Putting theories together to make specifications. In: Reddy, R. (ed.) Proceedings of 5th International Joint Conference on Artificial Intelligence, Cambridge, MA, pp. 1045–1058 (1977)

    Google Scholar 

  18. Calligaris, P., Omodeo, E.G., Tomescu, A.I.: A proof-checking experiment on representing graphs as membership digraphs. In: Cantone, D., Asmundo, M.N. (eds.) CILC 2013, 28th Italian Conference on Computational Logic, CEUR Workshop Proceedings, Catania, pp. 227–233 (2013)

    Google Scholar 

  19. Cantone, D., Omodeo, E.G., Policriti, A.: Set Theory for Computing. From Decision Procedures to Declarative Programming with Sets. Texts and Monographs in Computer Science. Springer, New York (2001)

    Book  MATH  Google Scholar 

  20. Cantone, D., Omodeo, E.G., Schwartz, J.T., Ursino, P.: Notes from the logbook of a proof-checker’s project. In: Dershowitz, N. (ed.) Verification: Theory and Practice, Essays Dedicated to Zohar Manna on the Occasion of his 64th Birthday. Lecture Notes in Computer Science, vol. 2772, pp. 182–207. Springer (2003)

    Google Scholar 

  21. Casagrande, A., Omodeo, E.G.: Reasoning about connectivity without paths. In: Bistarelli, S., Formisano, A. (eds.) ICTCS’13, Fifteenth Italian Conference on Theoretical Computer Science, CEUR Workshop Proceedings, pp. 93–108 (2014)

    Google Scholar 

  22. Casagrande, A., Piazza, C., Policriti, A.: Is hyper-extensionality preservable under deletions of graph elements? Electron. Notes Theor. Comput. Sci. 322, 103–118 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164 (1), 51–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cohen, P.J.: Set Theory and the Continuum Hypothesis. Mathematics Lecture Note Series. W. A. Benjamin, Inc., Reading, MA(1966)

    MATH  Google Scholar 

  25. D’Agostino, G., Omodeo, E.G., Policriti, A., Tomescu, A.I.: Mapping sets and hypersets into numbers. Fundam. Inform. 140 (3–4), 307–328 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Davis, M.: Applied Nonstandard Analysis. John Wiley & Sons, New York (1977)

    MATH  Google Scholar 

  27. Davis, M., Schonberg, E. (eds.): From Linear Operators to Computational Biology: Essays in Memory of Jacob T. Schwartz. Springer (2012)

    Google Scholar 

  28. Dewar, R.: SETL and the Evolution of Programming. In: [27, pp. 39–46] (2012)

    Google Scholar 

  29. Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: {log}: A language for programming in logic with finite sets. J. Log. Program. 28 (1), 1–44 (1996)

    Google Scholar 

  30. Dovier, A., Piazza, C., Policriti, A.: An efficient algorithm for computing bisimulation equivalence. Theor. Comput. Sci. 311 (1–3), 221–256 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dovier, A., Piazza, C., Pontelli, E., Rossi, G.: Sets and constraint logic programming. ACM Trans. Program. Lang. Syst. 22 (5), 861–931 (2000)

    Article  Google Scholar 

  32. Dyer, M., Frieze, A., Jerrum, M.: Approximately counting Hamilton paths and cycles in dense graphs. SIAM J. Comput. 27 (5), 1262–1272 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Hartcourt/Academic Press, Burlington (2001)

    MATH  Google Scholar 

  34. Erdős, P., Moser, L.: On the representation of directed graphs as unions of orderings. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 9, 125–132 (1964)

    MathSciNet  MATH  Google Scholar 

  35. Euler, L.: Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae 8, 128–140 (1741)

    Google Scholar 

  36. Felgner, U.: Comparison of the axioms of local and universal choice. Fundamenta mathematicae 71 (1), 43–62 (1971)

    MathSciNet  MATH  Google Scholar 

  37. Ferro, A., Omodeo, E., Schwartz, J.: Decision procedures for some fragments of set theory. In: Bibel, W., Kowalski, R. (eds.) Proceedings of the 5th Conference on Automated Deduction. LNCS, vol. 87, pp. 88–96. Springer, Berlin/New York (1980)

    Google Scholar 

  38. Ferro, A., Omodeo, E.G., Schwartz, J.T.: Decision procedures for elementary sublanguages of set theory. I. Multi-level syllogistic and some extensions. Commun. Pure Applied Math. 33 (5), 599–608 (1980)

    Article  MATH  Google Scholar 

  39. Formisano, A., Omodeo, E.G.: Theory-specific automated reasoning. In: Dovier, A., Pontelli, E. (eds.) A 25-Year Perspective on Logic Programming: Achievements of the Italian Association for Logic Programming, GULP. Lecture Notes in Computer Science, vol. 6125, pp. 37–63. Springer, Berlin/Heidelberg (2010)

    Chapter  Google Scholar 

  40. Forti, M., Honsell, F.: Set theory with free construction principles. Annali Scuola Normale Superiore di Pisa, Classe di Scienze IV (10), 493–522 (1983)

    Google Scholar 

  41. Fraenkel, A.A.: The notion of “definite” and the independence of the axiom of choice. In: [46, pp. 284–289]. (The original publication, in German language, is of 1922)

    Google Scholar 

  42. Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)

    Google Scholar 

  43. Gentilini, R., Piazza, C., Policriti, A.: From bisimulation to simulation: Coarsest partition problems. J. Autom. Reason. 31 (1), 73–103 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gödel, K.: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis. Proc. Natl. Acad. Sci. U.S.A. 24 (12), 556–557 (1938)

    Article  MATH  Google Scholar 

  45. Harary, F., Palmer, E.: Graphical Enumeration. Academic Press, New York (1973)

    MATH  Google Scholar 

  46. van Heijenoort, J. (ed.): From Frege to Gödel – A Source Book in Mathematical Logic, 1879–1931, 3rd printing edn. Source books in the history of the sciences. Harvard University Press (1977)

    Google Scholar 

  47. Hendry, G., Vogler, W.: The square of a connected S(K 1, 3)-free graph is vertex pancyclic. J. Graph Theory 9 (4), 535–537 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hopcroft, J.E.: An nlogn algorithm for minimizing states in a finite automaton. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 189–196. Academic Press, New York (1971)

    Chapter  Google Scholar 

  49. Howorka, E.: On metric properties of certain clique graphs. J. Comb. Theory Ser. B 27 (1), 67–74 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  50. Jech, T.: Set Theory. Springer Monographs in Mathematics, 3rd Millennium edn. Springer, Berlin/Heidelberg (2003)

    Google Scholar 

  51. Keller, J.P., Paige, R.: Program derivation with verified transformations – a case study. Commun. Pure Appl. Math. 48 (9–10), 1053–1113 (1995). Special issue in honor of J.T. Schwartz

    Google Scholar 

  52. Kirby, L.: A hierarchy of hereditarily finite sets. Arch. Math. Log. 47 (2), 143–157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. König, D.: Theorie der endlichen und unendlichen graphen. Akademische Verlagsgesellschaft Leipzig (1936)

    Google Scholar 

  54. Kunen, K.: Set Theory. Studies in Logic. College Publications (2011)

    MATH  Google Scholar 

  55. Levin, D.A., Peresand, Y., Wilmer, E.: Markov Chains and Mixing Times. AMS, Providence (2009)

    Google Scholar 

  56. Lin, R., Olariu, S., Pruesse, G.: An optimal path cover algorithm for cographs. Comput. Math. Appl. 30 (8), 75–83 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. Lisitsa, A., Sazonov, V.: Linear ordering on graphs, anti-founded sets and polynomial time computability. Theor. Comput. Sci. 224, 173–213 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  58. Lovász, L., Plummer, M.D.: Matching Theory. North-Holland Mathematics Studies. North-Holland/Amsterdam, New York (1986). Includes indexes

    Google Scholar 

  59. Lozin, V.V., Milanič, M., Purcell, C.: Graphs without large apples and the maximum weight independent set problem. Graphs Comb. 30 (2), 395–410 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. Matthews, M.M., Sumner, D.P.: Hamiltonian Results in K 1, 3-Free Graphs. J. Graph Theory 8, 139–146 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  61. Melançon, G., Philippe, F.: Generating connected acyclic digraphs uniformly at random. Inf. Process. Lett. 90 (4), 209–213 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  62. Melançon, G., Dutour, I., Bousquet-Mélou, M.: Random generation of directed acyclic graphs. In: Nesetril, J., Noy, M., Serra, O. (eds.) Euroconference on Combinatorics, Graph Theory and Applications (Comb01). Electronic Notes in Discrete Mathematics, vol. 10, pp. 202–207 (2001)

    Article  MATH  Google Scholar 

  63. Milanič, M., Rizzi, R., Tomescu, A.I.: Set graphs. II. Complexity of set graph recognition and similar problems. Theor. Comput. Sci. 547, 70–81 (2014)

    MATH  Google Scholar 

  64. Milanič, M., Tomescu, A.I.: Set graphs. I. Hereditarily finite sets and extensional acyclic orientations. Discret. Appl. Math. 161 (4–5), 677–690 (2013)

    MATH  Google Scholar 

  65. Milanič, M., Tomescu, A.I.: Set graphs. IV. Further connections with claw-freeness. Discret. Appl. Math. 174, 113–121 (2014)

    MATH  Google Scholar 

  66. Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28 (3), 284–304 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  67. Mostowski, A.: An undecidable arithmetical statement. Fund. Math. 36, 143–164 (1949)

    MathSciNet  MATH  Google Scholar 

  68. von Neumann, J.: Zur Einführung der trasfiniten Zahlen. Acta Sci. Math. (Szeged) 1 (4–4), 199–208 (1922–23). Available in English translation in [46, pp. 346–354]

    Google Scholar 

  69. von Neumann, J.: Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre. J. für die reine und angewandte Mathematik (160), 227–241, reprinted in [70, pp. 494–508] (1929)

    Google Scholar 

  70. von Neumann, J.: Collected Works, vol. I: Logic, Theory of Sets and Quantum Mechanics. Pergamon Press, New York (1961)

    Google Scholar 

  71. Olariu, S.: The strong perfect graph conjecture for pan-free graphs. J. Comb. Theory Ser. B 47, 187–191 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  72. Omodeo, E.G., Cantone, D., Policriti, A., Schwartz, J.T.: A computerized referee. In: Stock, O., Schaerf, M. (eds.) Reasoning, Action and Interaction in AI Theories and Systems – Essays Dedicated to Luigia Carlucci Aiello. LNAI, vol. 4155, pp. 117–139. Springer (2006)

    Google Scholar 

  73. Omodeo, E.G., Parlamento, F., Policriti, A.: Decidability of ∃ ∗ ∀-sentences in membership theories. Math. Log. Q. 42 (1), 41–58 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  74. Omodeo, E.G., Policriti, A.: The Bernays-Schönfinkel-Ramsey class for set theory: semidecidability. J. Symb. Logic 75 (2), 459–480 (2010)

    Article  MATH  Google Scholar 

  75. Omodeo, E.G., Policriti, A.: The Bernays-Schönfinkel-Ramsey class for set theory: decidability. J. Symb. Logic 77 (3), 896–918 (2012)

    Article  MATH  Google Scholar 

  76. Omodeo, E.G., Policriti, A., Tomescu, A.I.: Statements of ill-founded infinity in set theory. In: Cégielski, P. (ed.) Studies in Weak Arithmetics. Lecture Notes, vol. 196, pp. 173–199. Center for the Study of Language and Information, Stanford University (2009)

    Google Scholar 

  77. Omodeo, E.G., Policriti, A., Tomescu, A.I.: Infinity, in short. J. Logic Comput. 22 (6), 1391–1403 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  78. Omodeo, E.G., Policriti, A., Tomescu, A.I.: Set-syllogistics meet combinatorics. Math. Struct. Comput. Sci. 27 (2), 296–310 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  79. Omodeo, E.G., Schwartz, J.T.: A ‘theory’ mechanism for a proof-verifier based on first-order set theory. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond, Essays in Honour of Robert A. Kowalski, Part II. Lecture Notes in Computer Science, vol. 2408, pp. 214–230. Springer (2002)

    Google Scholar 

  80. Omodeo, E.G., Tomescu, A.I.: Set graphs. III. Proof Pearl: claw-free graphs mirrored into transitive hereditarily finite sets. J. Automat. Reason. 52 (1), 1–29 (2014)

    MATH  Google Scholar 

  81. Omodeo, E.G., Tomescu, A.I.: Set graphs. V. On representing graphs as membership digraphs. J. Log. Comput. 25 (3), 899–919 (2015)

    MATH  Google Scholar 

  82. Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16 (6), 973–989 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  83. Paige, R., Tarjan, R.E., Bonic, R.: A linear time solution to the single function coarsest partition problem. Theoret. Comput. Sci. 40, 67–84 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  84. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  85. Parlamento, F., Policriti, A.: Decision procedures for elementary sublanguages of set theory. IX. Unsolvability of the decision problem for a restricted subclass of the  Δ 0-formulas in set theory. Commun. Pure Appl. Math. XLI, 221–251 (1988)

    Google Scholar 

  86. Parlamento, F., Policriti, A.: The logically simplest form of the infinity axiom. Proc. Am. Math. Soc. 103 (1), 274–276 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  87. Parlamento, F., Policriti, A.: Note on “The logically simplest form of the infinity axiom”. Proc. Am. Math. Soc. 108 (1), 285–286 (1990)

    MATH  Google Scholar 

  88. Parlamento, F., Policriti, A.: Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability. J. Automat. Reason. 7 (2), 271–284 (1991)

    Google Scholar 

  89. Parlamento, F., Policriti, A.: Expressing infinity without foundation. J. Symb. Logic 56 (4), 1230–1235 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  90. Parlamento, F., Policriti, A.: Undecidability results for restricted universally quantified formulae of set theory. Commun. Pure Appl. Math. XLVI (1), 57–73 (1993)

    Google Scholar 

  91. Paulson, L.C.: A simple formalization and proof for the mutilated chess board. Logic J. IGPL 9 (3), 499–509 (2001)

    Article  MathSciNet  Google Scholar 

  92. Peddicord, R.: The number of full sets with n elements. Proc. Am. Math. Soc. 13, 825–828 (1962)

    MathSciNet  MATH  Google Scholar 

  93. Piazza, C., Policriti, A.: Ackermann encoding, bisimulations, and OBDDs. Theory Pract. Logic Programm. 4 (5–6), 695–718 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  94. Policriti, A., Tomescu, A.I.: Counting extensional acyclic digraphs. Inf. Process. Lett. 111 (3), 787–791 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  95. Policriti, A., Tomescu, A.I.: Markov chain algorithms for generating sets uniformly at random. Ars Math. Contemp. 6 (57–68) (2013)

    Google Scholar 

  96. Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27, 742–744 (1918)

    MATH  Google Scholar 

  97. Quaife, A.: Automated deduction in von Neumann-Bernays-Gödel set theory. J. Autom. Reason. 8 (1), 91–147 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  98. Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930). Read on 13 Dec 1928

    Google Scholar 

  99. Rizzi, R., Tomescu, A.I.: Ranking, unranking and random generation of extensional acyclic digraphs. Inf. Process. Lett. 113 (5–6), 183–187 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  100. Robinson, R.M.: The theory of classes, a modification of von Neumann’s system. J. Symb. Logic 2, 29–36 (1937)

    Article  MATH  Google Scholar 

  101. Robinson, R.W.: Enumeration of acyclic digraphs. In: Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and its Applications. University of North Carolina, Chapel Hill (1970)

    MATH  Google Scholar 

  102. Robinson, R.W.: Counting labeled acyclic digraphs. In: Harary, F. (ed.) New Directions in the Theory of Graphs, pp. 239–273. Academic Press, New York (1973)

    Google Scholar 

  103. Sbihi, N.: Algorithme de recherche d’un stable de cardinalite maximum dans un graphe sans etoile. Discret. Math. 29 (1), 53–76 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  104. Schwartz, J., Dewar, R., Dubinsky, E., Schonberg, E.: Programming with Sets: An Introduction to SETL. Texts and Monographs in Computer Science. Springer, New York (1986)

    Book  MATH  Google Scholar 

  105. Schwartz, J.T., Cantone, D., Omodeo, E.G.: Computational Logic and Set Theory. Springer (2011). Foreword by Martin Davis

    Google Scholar 

  106. Steinsky, B.: Efficient coding of labeled directed acyclic graphs. Soft Comput. 7, 350–356 (2003)

    Article  MATH  Google Scholar 

  107. Steinsky, B.: Enumeration of labelled chain graphs and labelled essential directed acyclic graphs. Discret. Math. 270 (1–3), 266–277 (2003)

    MathSciNet  MATH  Google Scholar 

  108. Sumner, D.: Graphs with 1-factors. Proc. Am. Math. Soc. 42, 8–12 (1974)

    MATH  Google Scholar 

  109. Tarski, A.: Sur les ensembles fini. Fundamenta Mathematicae VI, 45–95 (1924)

    Google Scholar 

  110. Tomescu, A.I.: A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs. Discret. Math. 312 (15), 2388–2391 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  111. Vergnas, M.L.: A note on matchings in graphs. Cahiers Centre Etudes Rech. Opér. 17, 257–260 (1975)

    MathSciNet  MATH  Google Scholar 

  112. Wagner, S.: Asymptotic enumeration of extensional acyclic digraphs. In: Proceedings of the ANALCO12 Meeting on Analytic Algorithmics and Combinatorics, pp. 1–8 (2012)

    Google Scholar 

  113. West, D.B.: Introduction to Graph Theory. Prentice-Hall, New Jersey (2001)

    Google Scholar 

  114. Zermelo, E.: Untersuchungen über die Grundlagen der Mengenlehre I. In: [46, pp. 199–215] (1908)

    Google Scholar 

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Authors and Affiliations

  1. DMG/DMI (Department of Mathematics and Geosciences, Section of Studies in Mathematics and Information Technology), University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

  2. DMIF (Department of Mathematics Computer Science, and Physics), University of Udine, Udine, Italy

    Alberto Policriti

  3. Department of Computer Science Helsinki Institute for Information Technology HIIT, University of Helsinki, Helsinki, Finland

    Alexandru I. Tomescu

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  2. Alberto Policriti
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  3. Alexandru I. Tomescu
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Omodeo, E.G., Policriti, A., Tomescu, A.I. (2017). Infinite Sets and Finite Combinatorics. In: On Sets and Graphs. Springer, Cham. https://doi.org/10.1007/978-3-319-54981-1_8

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  • DOI: https://doi.org/10.1007/978-3-319-54981-1_8

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