Shadows and intersections: Stability and new proofs

Keevash, Peter

doi:10.1016/j.aim.2008.03.023

Cited by 57 publications

(62 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We shall frequently use the following inequality, whose simple proof forms part of Lemma 12 in [9]. Lemma 16.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Several algebraic versions of Lovász's result exist [2], [9]; however, no analogue of the full Kruskal-Katona theorem in linear algebra has so far been obtained.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we will be mainly concerned with the algebraic generalization due to Keevash [9]. To state it we will first introduce several definitions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the rank of higher inclusion matrices

Grosu¹,

Person²,

Szabó³

2014

Journal of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. Let r ≥ s ≥ 0 be integers and G be an r-graph. The higher inclusion matrix M r s (G) is a {0, 1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V (G) of size s: the entry corresponding to an edge e and a subset S is 1 if S ⊆ e and 0 otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function rex(n, t, r, s) as the maximum number of edges of an r-graph G having rk M r s (G) ≤ n s − t. For t at most linear in n we determine this function as well as the extremal r-graphs. The special case t = 1 answers a question of Keevash.

show abstract

“…We shall frequently use the following inequality, whose simple proof forms part of Lemma 12 in [9]. Lemma 16.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Several algebraic versions of Lovász's result exist [2], [9]; however, no analogue of the full Kruskal-Katona theorem in linear algebra has so far been obtained.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the rank of higher inclusion matrices

Grosu¹,

Person²,

Szabó³

2014

Journal of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Indeed, the families which are extremal for the argument in the proof of Lemma 3.4 must possess a great deal of structure. Instead of the Kruskal-Katona theorem, one should be able to use a stability version of the Kruskal-Katona theorem, as proved by Keevash [17] for example, to prove a more general result that accounts for the structure of the family under consideration.…”

Section: To See the Last Inequality That Is |E(g[a])|mentioning

confidence: 99%

Transference for the Erdős–ko–rado Theorem

2015

View full text Add to dashboard Cite

For natural numbers n, r ∈ N with n r , the Kneser graph K (n, r ) is the graph on the family of r -element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K (n, r ) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős-Ko-Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

show abstract

“…In the celebrated paper [1], Ahlswede and Khachatrian extended the Erdős-Ko-Rado theorem by determining the structure of all t-intersecting set systems of maximum size for all possible n (see also [3,17,25,29,37,39,40,41] for some related results). There have been many recent results showing that a version of the Erdős-Ko-Rado theorem holds for combinatorial objects other than set systems.…”

Section: Introductionmentioning

confidence: 99%

An Analogue of the Hilton-Milner Theorem for Weak Compositions

Ku¹,

Wong²

2015

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. Let N 0 be the set of non-negative integers, and let P (n, l) denote the set of all weak compositions of n with l parts, i.e., P (n, l) = {(x 1 , x 2 , . . . , x l ) ∈ N l 0 :A family A ⊆ P (n, l) is said to be trivially t-intersecting if there is a t-set T of [l] = {1, 2, . . . , l} and elements ys ∈ N 0 (s ∈ T ) such that A = {u ∈ P (n, l) : u(j) = y j for all j ∈ T }. We prove that given any positive integers l, t with l ≥ 2t + 3, there exists a constant n 0 (l, t) depending only on l and t, such that for all n ≥ n 0 (l, t), if A ⊆ P (n, l) is non-trivially t-intersecting, then Moreover, equality holds if and only if there is a t-set T of [l] such thatwhere As = {u ∈ P (n, l) : u(j) = 0 for all j ∈ T and u(s) = 0}and q i ∈ P (n, l) with q i (j) = 0 for all j ∈ [l] \ {i} and q i (i) = n.

show abstract

Shadows and intersections: Stability and new proofs

Cited by 57 publications

References 21 publications

On the rank of higher inclusion matrices

On the rank of higher inclusion matrices

Transference for the Erdős–ko–rado Theorem

An Analogue of the Hilton-Milner Theorem for Weak Compositions

Contact Info

Product

Resources

About