Turán number

In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by (Turán 1941), and the problem for general r was introduced in (Turán 1961). The paper (Sidorenko 1995) gives a survey of Turán numbers.

Definitions

Fix a set X of n vertices. For given r, an r-edge or block is a set of r vertices. A set of blocks is called a Turán (n,k,r) system (n ≥ k ≥ r) if every k-element subset of X contains a block. The Turán number T(n,k,r) is the minimum size of such a system.

Example

The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.

Relations to other combinatorial designs

It can be shown that

[math]\displaystyle{ T(n,k,r) \geq \binom{n}{r} {\binom{k}{r}}^{-1}. }[/math]

Equality holds if and only if there exists a Steiner system S(n - k, n - r, n).

An (n,r,k,r)-lotto design is an (n, k, r)-Turán system. Thus, T(n,k, r) = L(n,r,k,r).

See also

• Forbidden subgraph problem
• Combinatorial design

References

Bibliography

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8, https://archive.org/details/handbookofcombin0000unse 
• Hazewinkel, Michiel, ed. (2001), "Turán number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Sidorenko, A. (1995), "What we know and what we do not know about Turán numbers", Graphs and Combinatorics 11 (2): 179–199, doi:10.1007/BF01929486 
• Turán, P (1941), "Egy gráfelméleti szélsőértékfeladatról (Hungarian. An extremal problem in graph theory.)" (in Hungarian), Mat. Fiz. Lapok 48: 436–452 
• Turán, P. (1961), "Research problems", Magyar Tud. Akad. Mat. Kutato Int. Közl. 6: 417–423