• Corpus ID: 235391058

Best possible bounds on the number of distinct differences in intersecting families

@article{Frankl2021BestPB,
  title={Best possible bounds on the number of distinct differences in intersecting families},
  author={Peter Frankl and S. G. Kiselev and Andrey B. Kupavskii},
  journal={ArXiv},
  year={2021},
  volume={abs/2106.05355}
}
For a family F , let D(F) stand for the family of all sets that can be expressed as F ∖ G, where F,G ∈ F . A family F is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of |D(F)| for an intersecting family of k-element sets? Frankl conjectured that the maximum is attained when F is the family of all sets containing a fixed element. We show that this holds if n > 50k log k and k > 50. At the same time… 
Triangles in r-wise t-intersecting families
Let t , r , k and n be positive integers and F a family of k -subsets of an n -set V . The family F is r -wise t -intersecting if for any F 1 , . . . , F r ∈ F , we have |∩ ri =1 F i | > t . An r
Triangles in intersecting families
We prove the following the generalized Turán type result. A collection T of r sets is an r-triangle if for every T1, T2, . . . , Tr−1 ∈ T we have ∩ r−1 i=1 Ti 6= ∅, but ∩T∈T T is empty. A family F of

References

SHOWING 1-10 OF 14 REFERENCES
Intersecting Families are Essentially Contained in Juntas
TLDR
It is proved that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersectingfamily), and the methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.
Erdös-Ko-Rado theorem with conditions on the maximal degree
  • P. Frankl
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1987
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
A theorem of finite sets
HAJNAL proved this statement in the case of l = 3 (unpublished). In this paper I prove for all cases that this is, indeed, the minimum, and find the (more complicated) minimum also for arbitrary n.
Differences of sets and a problem of Graham
R. L. Graham has posed the following question: Given n positive integers a1, < a2 <… < an, does there exists a pair of indices i, j such that ai/(ai, aj) ⩾ n? ((ai, aj) = g.c.d. of ai and aj).
Combinatorial problems and exercises
Problems Hints Solutions Dictionary of the combinatorial phrases and concepts used Notation Index of the abbreviations of textbooks and monographs Subject index Author index Errata.
The number of simplices in a complex
  • Math. Optimization Techniques, pp. 251– 278, Univ. of Calif. Press, Berkeley
  • 1963
...
...