Abstract
The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties:
-
i)
ifS 1,S 2, ...,S k + 1 are distinct pariwise disjoint elements ofL then there exists nonemptyI ⊂ {1, 2, ...,k + 1} with ∪ i∈I S i ⋃L
-
ii)
ifL =L 1 ⋃...⋃L r is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS 1,S 2, ...,S k ∈L j , such thatL i∈I S i ∈L j for every nonemptyI ⊂ {1, 2, ...,k}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. New York: John Wiley & Sons 1980
Graham, R.L.: Rudiments of Ramsey Theory. Providence, RI: AMS 1981
Nešetřil, J., Rödl, V.: The Ramsey property for graphs with forbidden complete subgraphs. J. Comb. Theory (B)20, 243–249 (1976)
Nešetřil, J., Rödl, V.: Van der Waerden theorem for sequences of integers not containing arithmetical progression ofk terms. Commentat. Math. Univ. Carol.17, 675–681 (1976)
Nešetřil, J., Rödl, V.: Another proof of the Folkman-Rado-Sanders theorem, J. Comb. Theory (A)34, 108–109 (1983)
Rado, R.: Studien zur Kombinatorik. Math. Z.36, 242–280 (1933)
Ramsey, F.P.: On a problem of formal logic. Proc. London Math. Soc.30, 264–286 (1930)
Sanders, J.: A generalization of Schur's theorem (Dissertation, Yale University 1969)
Spencer, J.: Restricted Ramsey configurations. J. Comb. Theory (A)19, 275–286 (1975)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nešetřil, J., Rödl, V. Finite union theorem with restrictions. Graphs and Combinatorics 2, 357–361 (1986). https://doi.org/10.1007/BF01788110
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01788110