Extremal Combinatorics

Jukna, Stasys

doi:10.1007/978-3-642-17364-6

Cited by 127 publications

(16 citation statements)

References 120 publications

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“…Theorem 1.16 naturally fits into the wide literature on forbidden L-intersections in extremal set theory (see [2,4,19]). Here one aims to understand how large certain families of sets can be if all intersections between elements of A are restricted to lie in some set L. For example, the Erdős-Ko-Rado theorem [9] can be viewed as an L 0 -intersection theorem for families A ⊂ n k , where L 0 = {l ∈ N : 1 l k}.…”

Section: A Probabilistic Forbidden Intersection Theoremmentioning

confidence: 55%

“…Intersection theorems have been a central topic of Extremal Combinatorics since the seminal paper of Erdős, Ko and Rado [9], and the area has grown into a vast body of research (see [2,4,19] for an overview). The Frankl-Rödl forbidden intersection theorem is a fundamental result of this type, which has had a wide range of applications to different areas of mathematics, including discrete geometry [12], communication complexity [28] and quantum computing [6].…”

Section: Introductionmentioning

confidence: 99%

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Forbidden vector‐valued intersections

Keevash

Long

2020

Proc. London Math. Soc.

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We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl–Rödl forbidden intersection theorem in which set intersections are vector‐valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC‐dimension, dependent random choice and a new correlation inequality for product measures.

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Section: A Probabilistic Forbidden Intersection Theoremmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Forbidden vector‐valued intersections

Keevash

Long

2020

Proc. London Math. Soc.

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“…The corresponding result for real Hadamard matrices has been obtained by Alon (cf. [10], Lemma 14.6). Alon's proof can be trivially adapted to deal with complex Hadamard matrices.…”

Section: Hadamard Tradesmentioning

confidence: 96%

“…We have that ab n by Theorem 7. In the other direction, Lindsay's Lemma states that the size of a rank one submatrix of a Hadamard matrix of order n is bounded above by n (see Lemma 14.5 of [10]). …”

Section: Corollarymentioning

confidence: 99%

Trades in Complex Hadamard Matrices

Catháin

Wanless

2015

Springer Proceedings in Mathematics &Amp; Statistics

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A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c = 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.

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“…Note that the Combinatorial Nullstellensatz cannot be applied to any of the other monomial terms in f . After its discovery, the Combinatorial Nullstellensatz would soon become a powerful tool in extremal combinatorics [11]. With regards to graph labeling and coloring problems, it has been used to prove theorems on anti-magic labelings, neighbor sum distinguishing total colorings, and list colorings [28,36,9].…”

Section: The Combinatorial Nullstellensatzmentioning

confidence: 99%

Constructing integer-magic graphs via the Combinatorial Nullstellensatz

Low

Roberts

2022

Art Discrete Appl. Math.

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Let A be a nontrivial additive abelian group and A * = A \ {0}. A graph is A-magic if there exists an edge labeling f using elements of A * which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to construct nontrivial classes of Z p -magic graphs, prime p ≥ 3. For these graphs, some lower bounds on the number of distinct Z p -magic labelings are also established.

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Extremal Combinatorics

Cited by 127 publications

References 120 publications

Forbidden vector‐valued intersections

Forbidden vector‐valued intersections

Trades in Complex Hadamard Matrices

Constructing integer-magic graphs via the Combinatorial Nullstellensatz

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