On set systems not containing delta systems

Abbott, H. L.; Exoo, Geoffrey

doi:10.1007/bf01271703

Cited by 3 publications

(4 citation statements)

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“…Case [1,1]. If two of A, B and C belong to F 1 , then the intersection of these two has at least eight elements in common with W 1 _ W 2 .…”

Section: Outline Of the Constructionmentioning

confidence: 99%

“…So, we may assume A # F 1 , B # F 5 _ F 6 and C # F 7 _ F 8 . Moreover, we can assume B # Case [1,0]. We may assume that A and B are in F 1 and C # F 5 .…”

Section: Outline Of the Constructionmentioning

confidence: 99%

“…Erdo s and Rado [8] proved that (r&1) k < f (k, r)<k ! (r&1) k (1) and conjectured that for each r, there exists a constant C r so that f(k, r)<C k r . Erdo s (see [6]) has offered 1000 dollars for the proof or disproof of this for r=3.…”

Section: Introductionmentioning

confidence: 99%

“…As far as the lower bounds are concerned, limited progress seems to have been made since 1974 (see [1], [2], [4]). Infinite versions have also been studied in, for example, [7] and [9].…”

Section: Introductionmentioning

confidence: 99%

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Intersection Statements for Systems of Sets

Deuber¹,

Erdös

Gunderson³

et al. 1997

Journal of Combinatorial Theory, Series A

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A family of r sets is called a 2-system if any two sets have the same intersection. Denote by F(n, r) the most number of subsets of an n-element set which do not contain a 2-system consisting of r sets. Constructive new lower bounds for F(n, r) are given which improve known probabilistic results, and a new upper bound is given by employing an argument due to Erdo s and Szemere di. Another construction is given which shows that for certain n, F(n, 3) 1.551 n&2 . We also show a relationship between the upper bound for F(n, 3) and the Erdo s Rado conjecture on the largest uniform family of sets not containing a 2-system.

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“…Case [1,1]. If two of A, B and C belong to F 1 , then the intersection of these two has at least eight elements in common with W 1 _ W 2 .…”

Section: Outline Of the Constructionmentioning

confidence: 99%

“…So, we may assume A # F 1 , B # F 5 _ F 6 and C # F 7 _ F 8 . Moreover, we can assume B # Case [1,0]. We may assume that A and B are in F 1 and C # F 5 .…”

Section: Outline Of the Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As far as the lower bounds are concerned, limited progress seems to have been made since 1974 (see [1], [2], [4]). Infinite versions have also been studied in, for example, [7] and [9].…”

Section: Introductionmentioning

confidence: 99%

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