Generalized Paley graphs and their complete subgraphs of orders three and four

Abstract

Let \(k \ge 2\) be an integer. Let q be a prime power such that \(q \equiv 1 ({\mathrm{mod}}\,\,k)\) if q is even, or, \(q \equiv 1 ({\mathrm{mod}}\,\,2k)\) if q is odd. The generalized Paley graph of order q, \(G_k(q)\), is the graph with vertex set \(\mathbb {F}_q\) where ab is an edge if and only if \({a-b}\) is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in \(G_k(q)\), \(\mathcal {K}_4(G_k(q))\), which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original (\(k=2\)) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in \(G_k(q)\), \(\mathcal {K}_3(G_k(q))\). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of \(\mathcal {K}_4(G_k(q))\) (resp. \(\mathcal {K}_3(G_k(q))\)) yield lower bounds for the multicolor diagonal Ramsey numbers \(R_k(4)=R(4,4,\ldots ,4)\) (resp. \(R_k(3)\)). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both \(\mathcal {K}_4(G_k(q))\) and \(\mathcal {K}_3(G_k(q))\), when q is prime, and Fourier coefficients of modular forms.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ahlgren, S.: The points of a certain fivefold over finite fields and the twelfth power of the eta function. Finite Fields Appl. 8(1), 18–33 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ahlgren, S., Ono, K.: Modularity of a certain Calabi–Yau threefold. Monatsh. Math. 129(3), 177–190 (2000)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. Wiley, New York (1998)

    Google Scholar 

  4. 4.

    Chung, F.R.K.: On the Ramsey numbers N(3,3,3;2). Discrete Math. 5, 317–321 (1973)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Evans, R.J., Pulham, J.R., Sheehan, J.: On the number of complete subgraphs contained in certain graphs. J. Combin. Theory Ser. B 30(3), 364–371 (1981)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Frechette, S., Ono, K., Papanikolas, M.: Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Not. 60, 3233–3262 (2004)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fettes, S., Kramer, R., Radziszowski, S.P.: An upper bound of 62 on the classical Ramsey number R(3,3,3,3). Ars Combin. 72, 41–63 (2004)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Fuselier, J.G.: Traces of Hecke operators in level 1 and Gaussian hypergeometric functions. Proc. Am. Math. Soc. 141(6), 1871–1881 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gomez, A., McCarthy, D., Young, D.: Apéry-like numbers and families of newforms with complex multiplication. Res. Number Theory 5(1), Paper No. 5 (2019)

  10. 10.

    Greene, J.: Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields. Thesis (Ph.D.) University of Minnesota (1984)

  11. 11.

    Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77–101 (1987)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Greenwood, R.E., Gleason, A.M.: Combinatorial relations and chromatic graphs. Can. J. Math. 7, 1–7 (1955)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hill, R., Irving, R.W.: On group partitions associated with lower bounds for symmetric Ramsey numbers. Eur. J. Combin. 3, 35–50 (1982)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Jones, G.: Paley and the Paley graphs. arXiv:1702.00285v1

  15. 15.

    Lennon, C.: Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold. J. Number Theory 131(12), 2320–2351 (2011)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lim, T.K., Praeger, C.: On generalized Paley graphs and their automorphism groups. Mich. Math. J. 58(1), 293–308 (2009)

    MathSciNet  Article  Google Scholar 

  17. 17.

    LMFDB—The L-functions and Modular Forms Database. www.lmfdb.org

  18. 18.

    Martin, Y., Ono, K.: Eta-quotients and elliptic curves. Proc. Am. Math. Soc. 125(11), 3169–3176 (1997)

    MathSciNet  Article  Google Scholar 

  19. 19.

    McCarthy, D.: Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms. J. Number Theory 170, 263–281 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    McCarthy, D., Papanikolas, M.: A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform. Int. J. Number Theory 11(8), 2431–2450 (2015)

    MathSciNet  Article  Google Scholar 

  21. 21.

    McCarthy, D., Osburn, R., Straub, A.: Sequences, modular forms and cellular integrals. Math. Proc. Camb. Philos. Soc. 168(2), 379–404 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mortenson, E.: Supercongruences for truncated \({}_{n+1}F_{n}\) hypergeometric series with applications to certain weight three newforms. Proc. Am. Math. Soc. 133(2), 321–330 (2005)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ono, K.: A note on the Shimura correspondence and the Ramanujan \(\tau (n)\) function. Utilitas Math. 47, 153–160 (1995)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc. 350(3), 1205–1223 (1998)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)

    Article  Google Scholar 

  26. 26.

    Parnami, J.C., Agrawal, M.K., Rajwade, A.R.: Jacobi sums and cyclotomic numbers for a finite field. Acta Arith. 41(1), 1–13 (1982)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Katre, S.A., Rajwade, A.R.: Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum. Math. Scand. 60(1), 52–62 (1987)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Radziszowski, S.P.: Small Ramsey numbers. Electron. J. Combin. 1 (1994), Dynamic Survey 1

  29. 29.

    Xiaodong, X., Zheng, X., Exoo, G., Radziszowski, S.P.: Constructive lower bounds on classical multicolor Ramsey numbers. Electron. J. Combin. 11(1), Research Paper 35 (2004)

  30. 30.

    Zagier, D.: Elliptic Modular Forms and their Applications, The 1–2–3 of Modular Forms, 1–103, Universitext. Springer, Berlin (2008)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Madeline Locus Dawsey.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by an AMS-Simons travel grant from the American Mathematical Society and the Simons Foundation. The second author was supported by a grant from the Simons Foundation (#353329, Dermot McCarthy).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dawsey, M.L., McCarthy, D. Generalized Paley graphs and their complete subgraphs of orders three and four. Res Math Sci 8, 18 (2021). https://doi.org/10.1007/s40687-021-00254-7

Download citation

Mathematics Subject Classification

  • Primary 05C30
  • 11T24
  • Secondary 05C55
  • 11F11