Abstract
A variety of extremal problems are considered for the members of two particular families of simple 2-complexes. We let Bk denote the complex consisting of k 2-cells all sharing one single edge in common and Wk the complex consisting of k 2-cells sharing exactly one vertex in common. The problems treated here include the following questions for the Bk and the corresponding questions for the Wk.
Suppose that for each 2-coloring of the 2-cells of a complex G, either one color contains a monochromatic copy of Bk or the other contains a monochromatic copy of Bℓ. How many 2-cells must G have? How many vertices? How many vertices and how many 2-cells must G have if one of Bk and Bℓ must be "faithfully imbedded" in one of the colors? How do these numbers compare with the "generalized Ramsey numbers" for the pairs (Bk, Bℓ)? What is the smallest integer f(n, k) such that each 2-complex on n vertices having f(n, k) 2-cells contains a copy of Bk?
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
W. Brown, P. Erdős, and V. T. Sos, On the existence of triangulated spheres in 3-graphs and related problems. Period, Math. Hungar. 3(1973) 221–228.
J. Doyen, Constructions of disjoint Steiner triple systems. Proc. Amer. Math. Soc. 32(1972) 409–416.
R. A. Duke and F. Harary, Generalized Ramsey Theory VI: Ramsey numbers for small plexes. J. Austral. Math. Soc., to appear.
R. A. Duke, Ramsey numbers of families of 2-complexes. Proc. of the Sixth Southeastern Conference on Combinatories, Graph Theory, and Computing. Congressus Numerantium No. XIV, Utilitas Math., Winnipeg, Man. (1975) 265–277.
P. Erdős, Extremal problems on graphs and hypergraphs. Hypergraph Seminar. Springer-Verlag, New York (1974) 75–84.
P. Erdős, Problems and results on finite and infinite graphs, to appear.
P. Erdős, private communication.
P. Erdős, R. J. Faudree, R. H. Schelp and C. C. Rousseau, The size Ramsey number, to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duke, R.A. (1978). Some extremal problems for simple two-complexes. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070372
Download citation
DOI: https://doi.org/10.1007/BFb0070372
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08666-6
Online ISBN: 978-3-540-35912-8
eBook Packages: Springer Book Archive