Abstract
We study the limit probabilities of first-order properties for random graphs with vertices in a Boolean cube. We find sufficient conditions for a sequence of random graphs to obey the zero-one law for first-order formulas of bounded quantifier depth. We also find conditions implying a weakened version of the zero-one law.
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References
N. Alon and J. H. Spencer, The Probabilistic Method (JohnWiley & Sons, Hoboken, NJ, 2016).
J. Brenner and L. Cummings, “The Hadamard maximum determinant problem,” Amer. Math. Monthly 79, 626 (1972).
E. E. Demekhin, A. M. Raĭgorodskiĭ, and O. I. Rubanov, “Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size,” Mat. Sb. 204, 49 (2009) [Sb. Math. 204, 508 (2009)].
R. Fagin, “Probabilities on finite models,” J. Symbolic Logic 41, 50 (1976).
Yu. V. Glebskiĭ, D. I. Kogan, M. I. Liogon’kiĭ, and V. A. Talanov, “Range and degree of realizability of formulas in the restricted predicate calculus,” Kibernetika 5, 17 (1969) [Cybernetics 5, 142 (1969)].
E. S. Gorskaya, M. I. Mitricheva, V. Yu. Protasov, and A. M. Raĭgorodskiĭ, “Estimating the chromatic numbers of Euclidean space by convex minimization methods,” Mat. Sb. 200, 3 (2009) [Sb. Math. 200, 783 (2009)].
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam–London–New York–Tokyo, 2006).
S. N. Popova, “Zero-one law for random distance graphs with vertices in {-1, 0, 1}n,” Probl. Peredachi Inf. 50, 64 (2014) [Probl. Inf. Transm. 50, 57 (2014)].
A. M. Raĭgorodskiĭ, “Borsukš problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56, 107 (2001) [RussianMath. Surveys 56, 103 (2001)].
A. M. Raĭgorodskiĭ, “The Borsuk problem for integral polytopes,” Mat. Sb. 193, 139 (2002) [Sb. Math. 193, 1535 (2002)].
A. M. Raĭgorodskiĭ, “Borsuk and Hadwiger problems and systems of vectors with exclusions for scalar products,” UspekhiMat. Nauk 57, 159 (2002) [RussianMath. Surveys 57, 606 (2002)].
A. M. Raĭgorodskiĭ, “The Erdős–Hadwiger problem and the chromatic numbers of finite geometric graphs,” Mat. Sb. 196, 123 (2005) [Sb. Math. 196, 115 (2005)].
A. M. Raĭgorodskiĭ, “The Borsuk and Gru¨ nbaum problems for lattice polytopes,” Izv. Ross. Akad. Nauk, Ser. Mat. 69, 81 (2005) [Izv. Math. 69, 513 (2005)].
A. M. Raĭgorodskiĭ, Models of Random Graphs (MTsNMO, Moscow, 2011) [in Russian].
A. M. Raĭgorodskiĭand I. M. Shitova, “Chromatic numbers of real and rational spaces with real or rational forbidden distances,” Mat. Sb. 199, 107 (2008) [Sb. Math. 199, 579 (2008)].
S. Shelah and J. H. Spencer, “Zero-one laws for sparse random graphs,” J. Amer. Math. Soc. 1, 97 (1988).
N. K. Vereshchagin and A. Shen’, Lectures onMathematical Logic and the Theory of Algorithms. Part 2. Languages and Calculi (MTsNMO, Moscow, 2008) [in Russian].
M. E. Zhukovskiĭ, “The weak zero-one laws for the random distance graphs,” Dokl. Ross. Akad. Nauk 430, 314 (2010)[Dokl. Math. 81, 51 (2010)].
M. E. Zhukovskiĭ, “The weak zero-one laws for the random distance graphs,” Teor. Veroyatn. Primen. 55, 344 (2010) [Theor. Probab. Appl. 55, 356 (2011)].
M. E. Zhukovskiĭ, “On a sequence of random distance graphs subject to the zero-one law,” Probl. Peredachi Inf. 47, 39 (2011) [Probl. Inf. Transm. 47, 251 (2011)].
M. E. Zhukovskiĭ, “A weak zero-one law for sequences of random distance graphs,” Mat. Sb. 203, 95 (2012) [Sb. Math. 203, 1012 (2012)].
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Original Russian Text © S.N. Popova, 2016, published in Matematicheskie Trudy, 2016, Vol. 19, No. 1, pp. 106–177.
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Popova, S.N. Zero-one laws for random graphs with vertices in a Boolean cube. Sib. Adv. Math. 27, 26–75 (2017). https://doi.org/10.3103/S1055134417010035
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DOI: https://doi.org/10.3103/S1055134417010035