Abstract
There is a variety of problems in extremal combinatorics for which there is a unique configuration achieving the optimum value. Moreover, as the size of the problem grows, configurations that “almost achieve” the optimal value can be shown to be “almost equal” to the extremal configuration. This phenomenon, known as stability, has been formalized by Simonovits [A Method for Solving Extremal Problems in Graph Theory, Stability Problems, Theory of Graphs (Proc.Colloq., Tihany, 1966), 279–319] in the context of graphs, but has since been considered for several combinatorial structures. In this work, we describe a hypergraph extremal problem with an unusual combinatorial feature, namely, while the problem is unstable, it has a unique optimal solution up to isomorphism. To the best of our knowledge, this is the first such example in the context of (simple) hypergraphs.
More precisely, for fixed positive integers r and ℓ with 1 ≤ ℓ < r, and given an r-uniform hypergraph H, let κ(H, 4,ℓ) denote the number of 4-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ℓ elements. Consider the function KC\((n,r,4,\ell)=\max_{H\in{\mathcal H}_{n,r}} \kappa (H, 4,\ell) \), where the maximum runs over the family \({\mathcal H}_{n,r}\) of all r-uniform hypergraphs on n vertices. We show that, for n large, there is a unique n-vertex hypergraph H for which κ(H, 4,ℓ) = KC(n,r,4,ℓ), despite the fact that the problem of determining KC(n,r,4,ℓ) is unstable for every fixed r > ℓ ≥ 2.
This work was partially supported by the University of São Paulo, through the MaCLinC/NUMEC project.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahlswede, R., Khachatrian, L.H.: The complete intersection theorem for systems of finite sets. European J. Combin. 18(2), 125–136 (1997)
Blokhuis, A., Brouwer, A., Szőnyi, T., Weiner, Z.: On q-analogues and stability theorems. Journal of Geometry 101, 31–50 (2011)
Brown, W.G.: On an open problem of Paul Turán concerning 3-graphs. Studies in pure mathematics, pp. 91–93. Birkhäuser, Basel (1983)
Brown, W.G., Simonovits, M.: Extremal multigraph and digraph problems. Paul Erdős and his Mathematics, II (Budapest, 1999), Bolyai Soc. Math. Stud., János Bolyai Math. Soc. 11, 157–203 (2002)
Erdős, P.: Some recent results on extremal problems in graph theory. In: Results, Theory of Graphs, Internat. Sympos., Rome, pp. 117–123 (English) (1966), pp. 124–130 (French). Gordon and Breach, New York (1967)
Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2(12), 313–320 (1961)
Fon-Der-Flaass, D.G.: A method for constructing (3,4)-graphs. Mat. Zametki 44(4), 546–550, 559 (1988)
Frohmader, A.: More constructions for Turán’s (3,4)-conjecture. Electron. J. Combin. 15(1), Research Paper 137, 23 (2008)
Füredi, Z.: A new proof of the stability of extremal graphs: Simonovits’s stability from Szemerédi’s regularity. Talk at the Conference in Honor of the 70th Birthday of Endre Szemerédi, Budapest, August 2-7 (2010)
Füredi, Z.: Turán type problems, Surveys in combinatorics, Guildford. London Math. Soc. Lecture Note Ser., vol. 166, pp. 253–300. Cambridge Univ. Press, Cambridge (1991)
Füredi, Z., Simonovits, M.: Triple systems not containing a Fano configuration. Combin. Probab. Comput. 14(4), 467–484 (2005)
Hoppen, C., Kohayakawa, Y., Lefmann, H.: Edge colourings of graphs avoiding monochromatic matchings of a given size. Combin. Probab. Comput. 21(1-2), 203–218 (2012)
Hoppen, C., Kohayakawa, Y., Lefmann, H.: Hypergraphs with many Kneser colorings. European Journal of Combinatorics 33(5), 816–843 (2012)
Keevash, P.: Hypergraph Turán problems, Surveys in combinatorics 2011. London Math. Soc. Lecture Note Ser., vol. 392, pp. 83–140. Cambridge Univ. Press, Cambridge (2011)
Keevash, P., Mubayi, D.: Set systems without a simplex or a cluster. Combinatorica 30(2), 175–200 (2010)
Keevash, P., Sudakov, B.: The Turán number of the Fano plane. Combinatorica 25(5), 561–574 (2005)
Kostochka, A.V.: A class of constructions for Turán’s (3,4)-problem. Combinatorica 2(2), 187–192 (1982)
Simonovits, M.: A method for solving extremal problems in graph theory, stability problems. In: Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 279–319. Academic Press, New York (1968)
Simonovits, M.: Some of my favorite Erdős theorems and related results, theories. In: Paul Erdős and his Mathematics, II, Bolyai Soc. Math. Stud. (Budapest, 1999) János Bolyai Math. Soc. 11, 565–635 (2002)
Szőnyi, T., Weiner, Z.: On stability theorems in finite geometry, 38 pp. (2008) (preprint)
Turán, P.: Research problems. Magyar Tud. Akad. Mat. Kutató Int. Közl 6, 417–423 (1961)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hoppen, C., Kohayakawa, Y., Lefmann, H. (2013). An Unstable Hypergraph Problem with a Unique Optimal Solution. In: Aydinian, H., Cicalese, F., Deppe, C. (eds) Information Theory, Combinatorics, and Search Theory. Lecture Notes in Computer Science, vol 7777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36899-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-36899-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36898-1
Online ISBN: 978-3-642-36899-8
eBook Packages: Computer ScienceComputer Science (R0)