Turán Numbers for  3-Uniform Linear Paths of Length 3

Eliza Jackowska; Joanna Polcyn; Andrzej Ruciński

doi:10.37236/5320

Turán Numbers for 3-Uniform Linear Paths of Length 3

The Electronic Journal of Combinatorics ◽

10.37236/5320 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 3

Author(s):

Eliza Jackowska ◽

Joanna Polcyn ◽

Andrzej Ruciński

Keyword(s):

Exact Formula ◽

Uniform Hypergraph ◽

Linear Path ◽

Turán Number ◽

Analogous Formula

In this paper we confirm a special, remaining case of a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $\mathrm{ex}_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is not $C^3_3$-free.

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Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

Combinatorics Probability Computing ◽

10.1017/s0963548317000530 ◽

2017 ◽

Vol 27 (3) ◽

pp. 358-386 ◽

Cited By ~ 7

Author(s):

CLAYTON COLLIER-CARTAINO ◽

NATHAN GRABER ◽

TAO JIANG

Keyword(s):

Positive Integer ◽

Positive Constant ◽

Uniform Hypergraph ◽

Ramsey Numbers ◽

Turán Number ◽

Formula Group ◽

Image Position ◽

Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

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Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three

The Electronic Journal of Combinatorics ◽

10.37236/7119 ◽

2017 ◽

Vol 24 (3) ◽

Cited By ~ 2

Author(s):

Andrzej Ruciński ◽

Eliza Jackowska ◽

Joanna Polcyn

Keyword(s):

Ramsey Number ◽

Uniform Hypergraph ◽

Ramsey Numbers ◽

Turán Number ◽

Standard Application

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Turán-type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.

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A GENERALIZED EXACT FORMULA FOR THE SWIMMING COST OF UPSTREAM FISH MIGRATION

Journal of Japan Society of Civil Engineers Ser B1 (Hydraulic Engineering) ◽

10.2208/jscejhe.74.i_391 ◽

2018 ◽

Vol 74 (4) ◽

pp. I_391-I_396

Author(s):

Hidekazu YOSHIOKA ◽

Takeshi WATANABE ◽

Kentaro TSUGIHASHI

Keyword(s):

Exact Formula ◽

Fish Migration

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The Lower and Upper Bounds of Turán Number for Odd Wheels

Graphs and Combinatorics ◽

10.1007/s00373-021-02290-0 ◽

2021 ◽

Vol 37 (3) ◽

pp. 919-932

Author(s):

Byeong Moon Kim ◽

Byung Chul Song ◽

Woonjae Hwang

Keyword(s):

Upper Bounds ◽

Lower And Upper Bounds ◽

Turán Number

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Universal and unavoidable graphs

Combinatorics Probability Computing ◽

10.1017/s0963548321000110 ◽

2021 ◽

pp. 1-14

Author(s):

Matija Bucić ◽

Nemanja Draganić ◽

Benny Sudakov

Keyword(s):

Free Graph ◽

Turán Number ◽

Work Done

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

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Counting Hamilton cycles in Dirac hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000619 ◽

2020 ◽

pp. 1-23

Author(s):

Stefan Glock ◽

Stephen Gould ◽

Felix Joos ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Hamilton Cycle ◽

Uniform Hypergraph ◽

Similar Estimate ◽

Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

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New limits on tanβ for 2HDMs with Z2 symmetry

International Journal of Modern Physics A ◽

10.1142/s0217751x15501584 ◽

2015 ◽

Vol 30 (26) ◽

pp. 1550158 ◽

Cited By ~ 12

Author(s):

Dipankar Das

Keyword(s):

Lower Limit ◽

Scalar Potential ◽

Higgs Doublet ◽

Exact Formula ◽

Meson Mass ◽

Neutral Meson ◽

Breaking Parameter ◽

Two Higgs Doublet Models ◽

Breaking Term

In two-Higgs-doublet models with exact [Formula: see text] symmetry, putting [Formula: see text] at the alignment limit, the following limits on the heavy scalar masses are obtained from the conditions of unitarity and stability of the scalar potential: [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. The constraints from [Formula: see text] and neutral meson mass differences, when superimposed on the unitarity constraints, put a tighter lower limit on [Formula: see text] depending on [Formula: see text]. It has also been shown that larger values of [Formula: see text] can be allowed by introducing soft breaking term in the potential at the expense of a correlation between [Formula: see text] and the soft breaking parameter.

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Exact tensor closures for the three-dimensional Jeffery's equation

Journal of Fluid Mechanics ◽

10.1017/jfm.2011.165 ◽

2011 ◽

Vol 680 ◽

pp. 321-335 ◽

Cited By ~ 20

Author(s):

STEPHEN MONTGOMERY-SMITH ◽

WEI HE ◽

DAVID A. JACK ◽

DOUGLAS E. SMITH

Keyword(s):

Fibre Orientation ◽

Elliptic Integral ◽

Moment Tensor ◽

Three Dimensional ◽

Exact Formula ◽

Fourth Moment ◽

Newtonian Flow ◽

Curve Fit ◽

Explicit Formulae ◽

Jeffery’S Equation

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

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