An ordered version of the Erdös-Ko-Rado theorem

Livingston, Marilynn

doi:10.1016/0097-3165(79)90065-7

Cited by 32 publications

(19 citation statements)

References 3 publications

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“…This generalises a well-known result that was first stated by Meyer [31] and proved in different ways by Deza and Frankl [10], Bollobás and Leader [4], Engel [11] and Erdős et al [12], and that can be described by saying that the conjecture is true for F = [n] r . Berge [3] and Livingston [30] had proved (i) and (ii) respectively for the special case F = {[n]} (other proofs are found in [18,32]). In [5] the conjecture is also verified for F uniform and EKR; Holroyd and Talbot [20] had essentially proved (i) for such a family F in a graph-theoretical context.…”

Section: Intersecting Families Of Signed Setsmentioning

confidence: 95%

On t-intersecting families of signed sets and permutations

Borg

2009

Discrete Mathematics

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Section: Intersecting Families Of Signed Setsmentioning

confidence: 95%

On t-intersecting families of signed sets and permutations

Borg

2009

Discrete Mathematics

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“…There are several other papers in the general area, for example [1][2][3]7,[10][11][12]14,[16][17][18]24,27].…”

Section: Theorem 13 (See Deza and Franklmentioning

confidence: 98%

Multiple cross-intersecting families of signed sets

Borg¹,

Leader²

2010

Journal of Combinatorial Theory, Series A

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A k-signed r-set on [n] = {1, . . . ,n} is an ordered pair (A, f ), where A is an r-subset of [n] and f is a function from A to [k]. Families A 1 , . . . , A p are said to be cross-intersecting if any set in any family A i intersects any set in any other family A j . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.

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“…When m = n, we get the well known EKR Theorem. When m = 0, Berge [1] determined the maximum size of intersecting families of labeled n-sets, Livingston [11] characterized partial optimal intersecting families and Borg [3] completely solved it by using the shift operator in an inductive argument. Theorem 1.2 [1,3,11] .…”

Section: Introductionmentioning

confidence: 99%

Erdős-Ko-Rado theorems of labeled sets

Geng

2011

Acta Math. Appl. Sin. Engl. Ser.

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For k = (k 1 , · · · , kn) ∈ N n , 1 ≤ k 1 ≤ · · · ≤ kn, let L r k be the family of labeled r-sets on k given by L r k := {{(a 1 , la 1 ), · · · , (ar , la r )} : {a 1 , · · · , ar} ⊆ [n], la i ∈ [ka i ], i = 1, · · · , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.

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An ordered version of the Erdös-Ko-Rado theorem

Cited by 32 publications

References 3 publications

On t-intersecting families of signed sets and permutations

On t-intersecting families of signed sets and permutations

Multiple cross-intersecting families of signed sets

Erdős-Ko-Rado theorems of labeled sets

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