Tilted Sperner families

Leader, Imre; Long, Eoin

doi:10.1016/j.dam.2012.02.024

Cited by 8 publications

(23 citation statements)

References 3 publications

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“…Given

A, B \subseteq [n]

the subcube of

P (n)

spanned by A and B consists of all subsets of

A \cup B

that contain

A \cap B

. Kalai (see ) observed that

scriptA

is an antichain precisely if it does not contain A and B such that, in the subcube of

P (n)

spanned by A and B, A is the top point and B is the bottom point. He asked what happens if one ‘tilts’ this condition.…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

“…Let

p, q \in ℕ

be coprime with p < q . Leader and Long proved that the largest ( p, q )‐tilted Sperner family in

P (n)

has size

(q - p + o (1)) true(\begin{matrix} n \\ n / 2 \end{matrix} true)

, where the lower bound is obtained by considering the union of the q – p middle layers of the Boolean lattice (see for an explanation of this).…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

“…To prove Theorem 4.1 we will apply the following supersaturation version of the Leader–Long theorem . The proof applies the same averaging argument strategy used in . Lemma Let

p, q \in ℕ

be coprime with p < q.…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

“…k is an antichain for any 0 ≤ k ≤ n. A celebrated theorem of Sperner [41] states that in fact no antichain in P(n) has size larger than n n/2 . Given A, B ⊆ [n] the subcube of P(n) spanned by A and B consists of all subsets of A ∪ B that contain A ∩ B. Kalai (see [32]) observed that A is an antichain precisely if it does not contain A and B such that, in the subcube of P(n) spanned by A and B, A is the top point and B is the bottom point. He asked what happens if one 'tilts' this condition.…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

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Applications of graph containers in the Boolean lattice

Balogh

Treglown

Wagner

2016

Random Struct Algorithms

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We apply the graph container method to prove a number of counting results for the Boolean lattice P(n). In particular, we: Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in P(n), and we also give an upper bound on the number of transportation codes; Provide an alternative proof of Kleitman's theorem on the number of antichains in P(n) and give a two‐coloured analogue; Give an asymptotic formula for the number of (p, q)‐tilted Sperner families in P(n); Prove a random version of Katona's t‐intersection theorem. In each case, to apply the container method, we first prove corresponding supersaturation results. We also give a construction which disproves two conjectures of Ilinca and Kahn on maximal independent sets and antichains in the Boolean lattice. A number of open questions are also given. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 845–872, 2016

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“…Given

A, B \subseteq [n]

the subcube of

P (n)

spanned by A and B consists of all subsets of

A \cup B

that contain

A \cap B

. Kalai (see ) observed that

scriptA

is an antichain precisely if it does not contain A and B such that, in the subcube of

P (n)

spanned by A and B, A is the top point and B is the bottom point. He asked what happens if one ‘tilts’ this condition.…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

“…Let

p, q \in ℕ

be coprime with p < q . Leader and Long proved that the largest ( p, q )‐tilted Sperner family in

P (n)

has size

(q - p + o (1)) true(\begin{matrix} n \\ n / 2 \end{matrix} true)

, where the lower bound is obtained by considering the union of the q – p middle layers of the Boolean lattice (see for an explanation of this).…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

“…To prove Theorem 4.1 we will apply the following supersaturation version of the Leader–Long theorem . The proof applies the same averaging argument strategy used in . Lemma Let

p, q \in ℕ

be coprime with p < q.…”

Section: Tilted Sperner Familiesmentioning

confidence: 99%

Section: Tilted Sperner Familiesmentioning

confidence: 99%

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Applications of graph containers in the Boolean lattice

Balogh

Treglown

Wagner

2016

Random Struct Algorithms

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“…Kalai raised the question of how large a tilted Sperner family A ⊂ P[n] can be. In [13], Leader and the second author proved that such families satisfy |A| ≤ (1 + o(1)) n n/2 , which is asymptotically optimal. For sufficiently large n, the extremal families were also determined.…”

Section: Introductionmentioning

confidence: 99%

Set families with a forbidden pattern

Karpas

Long

2017

European Journal of Combinatorics

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A balanced pattern of order 2d is an element P ∈ {+, −} 2d , where both signs appear d times. Two sets A, B ⊂ [n] form a P -pattern, which we denote by pat(A,We consider the following extremal question: how large can a family A ⊂ P[n] be if A is P -free?We prove a number of results on the sizes of such families. In particular, we show that for some fixed c > 0, if P is a d-balanced pattern with d < c log log n then |A| = o(2 n ). We then give stronger bounds in the cases when (i) P consists of d + signs, followed by d − signs and (ii) P consists of alternating signs. In both cases, if d = o( √ n) then |A| = o(2 n ). In the case of (i), this is tight.

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Forbidding a set difference of size 1

Leader¹,

Long

2014

Discrete Applied Mathematics

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How large can a family A ⊂ P[n] be if it does not contain A, B with |A \ B| = 1? Our aim in this paper is to show that any such family has size at most 2+o (1) n n ⌊n/2⌋ . This is tight up to a multiplicative constant of 2. We also obtain similar results for families A ⊂ P[n] with |A \ B| = k, showing that they satisfy |A| ≤ C k n k n ⌊n/2⌋ , where C k is a constant depending only on k.

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Tilted Sperner families

Abstract: Let A be a family of subsets of an n-set such that A does not contain distinct sets A and B with |A\B| = 2|B\A|. How large can A be? Our aim in this note is to determine the maximum size of such an A. This answers a question of Kalai. We also give some related results and conjectures.

Cited by 8 publications

References 3 publications

Applications of graph containers in the Boolean lattice

Applications of graph containers in the Boolean lattice

Set families with a forbidden pattern

Forbidding a set difference of size 1

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