The Mathematics of Paul Erdös II pp 229-235 | Cite as

# A Bound of the Cardinality of Families not Containing Δ-Systems

Chapter

## Summary

P. Erdős and R. Rado defined a *Δ*-system as a family in which every two members have the same intersection. Here we obtain a new upper bound of the maximum cardinality *φ*(*n*) of an *n*-uniform family not containing any *Δ*-system of cardinality 3. Namely, we prove that for any *α* > 1, there exists *C = C(α)* such that for any *n*, *φ*(*n*) ≤ *C n*!α^{−n }.

## Keywords

Greedy Algorithm Extremal Problem Minimum Cardinality Maximum Cardinality Preliminary Lemma
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## References

- 1.P.Erdős, Problems and results on set systems and hypergraphs,
*Extended Abstrect, Conf.on Extremal Problems for Finite Sets, 1991, Visegrad, Hungary*1991, 85–92.Google Scholar - 2.P.Erdős and R.Rado, Intersection theorems for systems of sets,
*J.London Math. Soc*. 35(1960), 85–90.MathSciNetCrossRefGoogle Scholar - 3.J.Spencer, Intersection theorems for systems of sets,
*Canad. Math. Bull*. 20(1977), 249–254.MathSciNetzbMATHCrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1997