A note on Huang–Zhao theorem on intersecting families with large minimum degree

Peter Frankl; Norihide Tokushige

doi:10.1016/j.disc.2016.10.011

On the ratio of maximum and minimum degree in maximal intersecting families

Discrete Mathematics ◽

10.1016/j.disc.2012.10.007 ◽

2013 ◽

Vol 313 (2) ◽

pp. 207-211

Author(s):

Zoltán Lóránt Nagy ◽

Lale Özkahya ◽

Balázs Patkós ◽

Máté Vizer

Keyword(s):

Minimum Degree ◽

Intersecting Families ◽

Maximum And Minimum Degree

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The Minimum Number of Nonnegative Edges in Hypergraphs

The Electronic Journal of Combinatorics ◽

10.37236/4402 ◽

2014 ◽

Vol 21 (3) ◽

Cited By ~ 4

Author(s):

Hao Huang ◽

Benny Sudakov

Keyword(s):

Vector Space ◽

Minimum Degree ◽

Constant Factor ◽

Total Weight ◽

Real Numbers ◽

Minimum Number ◽

Intersecting Families

An $r$-uniform $n$-vertex hypergraph $H$ is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of $H$. In this paper we show that for $n>10r^3$, every $r$-uniform $n$-vertex hypergraph with equal codegrees has the MMS property, and the bound on $n$ is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of $n>10k^3$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative $k$-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for $n \ge 4k$ and any weighting on the $1$-dimensional subspaces of $\mathbb{F}_{q}^n$ with nonnegative sum, the number of nonnegative $k$-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erdős-Ko-Rado theorem on $k$-intersecting families.

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Weighted multiply intersecting families

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.40.2003.3.2 ◽

2003 ◽

Vol 40 (3) ◽

pp. 287-291 ◽

Cited By ~ 1

Author(s):

Peter Frankl ◽

Norihide Tokushige

Keyword(s):

Intersecting Families

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The Order of Monochromatic Subgraphs with a Given Minimum Degree

The Electronic Journal of Combinatorics ◽

10.37236/1725 ◽

2003 ◽

Vol 10 (1) ◽

Author(s):

Yair Caro ◽

Raphael Yuster

Keyword(s):

Positive Integer ◽

Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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A simple removal lemma for large nearly-intersecting families

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2015.06.015 ◽

2015 ◽

Vol 49 ◽

pp. 93-99 ◽

Cited By ~ 4

Author(s):

Tuan Tran ◽

Shagnik Das

Keyword(s):

Intersecting Families ◽

Removal Lemma

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Matchings in hypergraphs of large minimum degree

Journal of Graph Theory ◽

10.1002/jgt.20139 ◽

2006 ◽

Vol 51 (4) ◽

pp. 269-280 ◽

Cited By ~ 45

Author(s):

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Minimum Degree

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The Performance of Secondary Nasal Alar Base Revision for Unilateral Cleft Lip by Single YV-Plasty (the Importance of Overcorrection During Surgery)

The Cleft Palate-Craniofacial Journal ◽

10.1177/10556656211010609 ◽

2021 ◽

pp. 105566562110106

Author(s):

Yoshitaka Matsuura ◽

Hideaki Kishimoto

Keyword(s):

Cleft Lip ◽

Minimum Degree ◽

Nasal Deformity ◽

Simple Method ◽

Primary Surgery ◽

Base Revision ◽

Alar Base ◽

Anterior Nasal Spine ◽

Nasal Floor ◽

Case Basis

Although primary surgery for cleft lip has improved over time, the degree of secondary cleft or nasal deformity reportedly varies from a minimum degree to a remarkable degree. Patients with cleft often worry about residual nose deformity, such as a displaced columella, a broad nasal floor, and a deviation of the alar base on the cleft side. Some of the factors that occur in association with secondary cleft or nasal deformity include a deviation of the anterior nasal spine, a deflected septum, a deficiency of the orbicularis muscle, and a lack of bone underlying the nose. Secondary cleft and nasal deformity can result from incomplete muscle repair at the primary cleft operation. Therefore, surgeons should manage patients individually and deal with various deformities by performing appropriate surgery on a case-by-case basis. In this report, we applied the simple method of single VY-plasty on the nasal floor to a patient with unilateral cleft to revise the alar base on the cleft side. We adopted this approach to achieve overcorrection on the cleft side during surgery, which helped maintain the appropriate position of the alar base and ultimately balanced the nose foramen at 13 months after the operation. It was also possible to complement the height of the nasal floor without a bone graft. We believe that this approach will prove useful for managing cases with a broad and low nasal floor, thereby enabling the reconstruction of a well-balanced nose.

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