• Corpus ID: 248496863

Intersections and Distinct Intersections in Cross-intersecting Families

@inproceedings{Frankl2022IntersectionsAD,
  title={Intersections and Distinct Intersections in Cross-intersecting Families},
  author={Peter Frankl and Jian Wang},
  year={2022}
}
Let F , G be two cross-intersecting families of k -subsets of { 1 , 2 , . . ., n } . Let F ∧ G , I ( F , G ) denote the families of all intersections F ∩ G with F ∈ F , G ∈ G , and all distinct intersections F ∩ G with F 6 = G, F ∈ F , G ∈ G , respectively. For a fixed T ⊂ { 1 , 2 , . . ., n } , let S T be the family of all k -subsets of { 1 , 2 , . . ., n } containing T . In the present paper, we show that |F ∧G| is maximized when F = G = S { 1 } for n ≥ 2 k 2 +8 k , while surprisingly |I ( F… 
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References

SHOWING 1-9 OF 9 REFERENCES

On the maximum number of distinct intersections in an intersecting family

A new generalization of the Erdős-Ko-Rado theorem

TLDR
This paper shows that the Erdős-Ko-Rado theorem holds forn>n0(k) if and only ifs≧2k, and sharpens a theorem of Bollobás.

INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

2. Notation The letters a, b, c, d, x, y, z denote finite sets of non-negative integers, all other lower-case letters denote non-negative integers. If fc I, then [k, I) denotes the set

Improved bounds for Erdős' Matching Conjecture

  • P. Frankl
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2013

Ein Satz über Untermengen einer endlichen

  • Menger, Math. Zeitschrift
  • 1928

SOME INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

The shifting technique in extremal set theory

  • 1987

The Erdős-Ko-Rado theorem is true for n = ckt

  • Coll. Math. Soc. J. Bolyai
  • 1978

The shifting technique in extremal set theory, Surveys in Combinatorics

  • 1987