Abstract
For every fixed integers r, s satisfying 2 ≤ r < s there exists some ε = ε(r, s) > 0 for which we construct explicitly an infinite family of graphs H r,s,n , where H r,s,n has n vertices, contains no clique on s vertices and every subset of at least n 1-ε of its vertices contains a clique of size r. The constructions are based on spectral and geometric techniques, some properties of Finite Geometries and certain isoperimetric inequalities.
Similar content being viewed by others
References
Alon, N.: Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6, 207–219 (1986)
Alon, N.: Tough Ramsey graphs without short cycles. J. Algebraic Combinatorics 4, 189–195 (1995)
Alon, N.: Explicit Ramsey graphs and orthonormal labelings. The Electronic J. Combinatorics 1, R12 (1994)
Alon, N., Boppana, R.B., Spencer, J.H.: An asymptotic isoperimetric inequality (in preparation)
Alon, N., Chung, F.R.K.: Explicit construction of linear sized tolerant networks. Discrete Math. 72, 15–19 (1988)
Alon, N., Spencer, J.H., The probabilistic method. New York: Wiley 1992
Bollobás, B., Random graphs. London: Academic Press 1985
Bollobás, B., Hind, H.R.: Graphs without large triangle-free subgraphs. Discrete Math. 87, 119–131 (1991)
Chung, F.R.K., Cleve, R., Dagum, P.: A note on constructive lower bounds for the Ramsey numbers R(3, t). J. Comb. Theory Ser. B 57, 150–155 (1993)
Erdös, P.: On the construction of certain graphs. J. Comb. Theory 1, 149–153 (1966)
Erdös, P., Gallai, T.: On the minimal number of vertices representing the edges of a graph. Publ. Math. Inst. Hungar. Acad. Sci 6, 181–203 (1961)
Erdös, P., Rogers, C.A.: The construction of certain graphs. Canad. J. Math. 14, 702–707 (1962)
Frankl, P., Wilson, R.M.: Intersection theorems with geometric consequences. Combinatorica 1, 357–368 (1981)
Graham, R.L., Rothschild, B.L., Spencer, J.H., Ramsey theory (Second Edition). New York: Wiley 1990
Hall, M., Combinatorial theory (Second Edition). New York: Wiley 1986
Kleitman, D.J.: On a combinatorial problem of Erdös. J. Comb. Theory 1, 209–214 (1966)
Krivelevich, M.: K s-free graphs without large K r-free subgraphs. Comb., Prob, and Computing 3, 349–354 (1994)
Krivelevich, M.: Bounding Ramsey numbers through large deviation inequalities. Random Str. Alg. 7, 145–155 (1995)
McDiarmid, C.J.H.: On the method of bounded differences. In: J. Siemons: Surveys in combinatorics 1989, London Math. Society Lecture Notes Series 141, pp. 148–188. Cambridge Univ. Press, 1989
Milman, V.D., Schechtman, G., Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Mathematics 1200. Berlin Heidelberg New York: Springer 1986
Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81, 73–205 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences
Supported in part by a Charles Clore Fellowship
Rights and permissions
About this article
Cite this article
Alon, N., Krivelevich, M. Constructive Bounds for a Ramsey-Type Problem. Graphs and Combinatorics 13, 217–225 (1997). https://doi.org/10.1007/BF03352998
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03352998