On the number of homogeneous subgraphs of a graph

Abstract

Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log2 n) ≧G(n) ≧ exp (0.2275 log2 n), the main tool is a Ramsey—Turán type theorem.

We formulate a conjecture what supports Thomason’s conjecture\(\mathop {\lim }\limits_{k \to \infty } \) R(k, k)1/k = 2.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. [1]

    P. Erdős, On the combinatorial problems which I would most like to see solved,Combinatorica 1 (1981), 25–42.

    MathSciNet  Google Scholar 

  2. [2]

    P. Erdős, On the number of complete subgraphs contained in certain graphs,Magyar Tud. Akad. Mat. Kut. Int. Közl. VII.series A, (3) (1962), 459–464.

    Google Scholar 

  3. [3]

    P. Erdős, Some remarks on the theory of graphs,Bull. Amer. Math. Soc. 53 (1947), 292–294.

    MathSciNet  Article  Google Scholar 

  4. [4]

    P. Erdős andG. Szekeres, A combinatorial problem in geometry,Compositio Math. 2 (1935), 463–470.

    MathSciNet  Google Scholar 

  5. [5]

    P. Erdős andE. Szemerédi, On a Ramsey type theorem,Period. Math. Hung. 2 (1972), 295–299.

    Article  Google Scholar 

  6. [6]

    A. W. Goodman, On sets of acquintances and strangers at any party,Amer. Math. Monthly 66 (1959), 778–783.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    A. Thomason, On finite Ramsey numbers,Europ. J. Combinatorics 3 (1982), 263–273.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Székely, L.A. On the number of homogeneous subgraphs of a graph. Combinatorica 4, 363–372 (1984). https://doi.org/10.1007/BF02579149

Download citation

AMS subject classification (1980)

  • 05 A 15
  • 05 C 55