The Induced Removal Lemma in Sparse Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 153-162
Author(s):  
Shachar Sapir ◽  
Asaf Shapira
Keyword(s):  
General Result ◽  
Original Proof ◽  
Sparse Graphs ◽  
Vertex Graph ◽  
Removal Lemma ◽  
Special Case

AbstractThe induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.

scholarly journals Hereditary quasirandomness without regularity

2017 ◽  
Vol 164 (3) ◽  
pp. 385-399 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Alternative Proof ◽  
Regularity Lemma ◽  
Original Proof ◽  
Vertex Graph

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

scholarly journals NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC

2008 ◽  
Vol 24 (5) ◽  
pp. 1443-1455 ◽  
Author(s):  
James Davidson ◽  
Jan R. Magnus ◽  
Jan Wiegerinck
Keyword(s):  
General Result ◽  
Random Variables ◽  
Nonparametric Test ◽  
Limiting Distribution ◽  
Residue Theorem ◽  
Prove Theorem ◽  
General Bound ◽  
Result Theorem ◽  
Special Case ◽  
Cauchy’S Residue Theorem

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

scholarly journals Bounded Height in Families of Dynamical Systems

2017 ◽  
Vol 2019 (8) ◽  
pp. 2453-2482 ◽  
Author(s):  
Laura DeMarco ◽  
Dragos Ghioca ◽  
Holly Krieger ◽  
Khoa Dang Nguyen ◽  
Thomas Tucker ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
General Result ◽  
Algebraic Integer ◽  
Arithmetic Dynamics ◽  
Special Case

Abstract Let $a,b\in\overline{\mathbb{Q}}$ be such that exactly one of $a$ and $b$ is an algebraic integer, and let $f_t(z):=z^2+t$ be a family of polynomials parameterized by $t\in\overline{\mathbb{Q}}$. We prove that the set of all $t\in\overline{\mathbb{Q}}$ for which there exist $m,n\geq 0$ such that $f_t^m(a)=f_t^n(b)$ has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.

scholarly journals Fatou's Lemma and Lebesgue's convergence theorem for measures

2000 ◽  
Vol 13 (2) ◽  
pp. 137-146 ◽  
Author(s):  
Onésimo Hernández-Lerma ◽  
Jean B. Lasserre
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
General Result ◽  
Convergence Theorem ◽  
Convergence Theorems ◽  
Vector Lattices ◽  
Fatou’S Lemma ◽  
Dominated Convergence ◽  
Special Case ◽  
Fatou's Lemma

Analogues of Fatou's Lemma and Lebesgue's convergence theorems are established for ∫fdμn when {μn} is a sequence of measures. A “generalized” Dominated Convergence Theorem is also proved for the asymptotic behavior of ∫fndμn and the latter is shown to be a special case of a more general result established in vector lattices and related to the Dunford-Pettis property in Banach spaces.

scholarly journals A spanning bandwidth theorem in random graphs

2021 ◽  
pp. 1-31
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Jakob Schnitzer ◽  
Anusch Taraz
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Spanning Subgraph ◽  
Vertex Graph ◽  
Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

scholarly journals The Amazing Power of Dimensional Analysis: Quantifying Market Impact

2017 ◽  
Vol 03 (03n04) ◽  
pp. 1850004 ◽  
Author(s):  
Mathias Pohl ◽  
Alexander Ristig ◽  
Walter Schachermayer ◽  
Ludovic Tangpi
Keyword(s):  
Dimensional Analysis ◽  
Market Microstructure ◽  
General Result ◽  
Scaling Relations ◽  
Square Root ◽  
Similar Argument ◽  
Market Impact ◽  
Functional Relations ◽  
Special Case

This paper complements the inspiring work on dimensional analysis and market microstructure by Kyle and Obizhaeva (2017). Following closely these authors, our main result shows, by a similar argument as usually applied in physics, the following remarkable fact. If the market impact of a meta-order only depends on four well-defined and financially meaningful variables, and some obvious scaling relations as well as the assumption of leverage neutrality are satisfied, then there is only one possible form of this dependence. In particular, the market impact is proportional to the square-root of the size of the meta-order. This theorem can be regarded as a special case of a more general result of Kyle and Obizhaeva. These authors consider five variables which might have an influence on the size of the market impact. In this case, one finds a richer variety of possible functional relations which we precisely characterize. We also discuss the analogies to classical arguments from physics, such as the period of a pendulum.

Lifting Invariants of Projective Subschemes

2003 ◽  
Vol 14 (05) ◽  
pp. 529-539
Author(s):  
Ph. Ellia
Keyword(s):  
General Result ◽  
Closed Subscheme ◽  
Restriction Map ◽  
Space Curves ◽  
Special Case

The lifting invariants of a closed subscheme X ⊂ Pn are the numbers [Formula: see text], where H is a general hyperplane and where f is the restriction map. The lifting invariants measure the obstruction to lift hypersurfaces (of H) containing X ∩ H to hypersurfaces containing X. We first prove a general result (which holds for every X ⊂ Pn ) on the behaviour of the ri's; then we turn to the special case of space curves and, under some special assumptions, we prove vanishing results for the ri's and for the cohomology.

scholarly journals The zeros ofaz2J″ν(z)+bzJ′ν(z)+cJν(z)as functions of order

1992 ◽  
Vol 15 (2) ◽  
pp. 319-322 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Bessel Function ◽  
General Result ◽  
Positive Zero ◽  
Special Case

Ifj″νkdenotes thekthpositive zero of the Bessel functionJ″ν(x), it has been shown recently by Lorch and Szego [2] thatj″ν1increases withνinν>0and that (withkfixed in2,3,…)j″νkincreases in0<ν≤3838. Furthermore, Wong and Lang have now extended the latter result, as well, to the rangeν>0. The present paper, by using a different kind of analysis, re-obtains these conclusions as a special case of a more general result concerning the positive zeros of the functionaz2J″ν(z)+bzJ′ν(z)+cJν(z). Here, the constantsa,bandcare subject to certain mild restrictions.

scholarly journals Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

10.37236/5340 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Peter Hegarty ◽  
Anders Martinsson
Keyword(s):  
Abelian Group ◽  
General Result ◽  
Arithmetic Progressions ◽  
Cyclic Groups ◽  
Special Case

A permutation $\pi$ of an abelian group $G$ is said to destroy arithmetic progressions (APs) if, whenever $(a, \, b, \, c)$ is a non-trivial 3-term AP in $G$, that is $c-b=b-a$ and $a, \, b, \, c$ are not all equal, then $(\pi(a), \, \pi(b), \pi(c))$ is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of $\mathbb{Z}_n$, for all $n \not\in \{2, \, 3, \, 5, \, 7\}$. Here we prove, as a special case of a more general result, that such a permutation exists for all $n \geq n_0$, for some explicitly constructed number $n_0 \approx 1.4 \times 10^{14}$. We also construct such a permutation of $\mathbb{Z}_p$ for all primes $p > 3$ such that $p \equiv 3 \; ({\hbox{mod $8$}})$.

scholarly journals Practical Compact Routing on Sparse Graphs

2021 ◽  
Author(s):  
Kai Hormann ◽  
Craig Gotsman
Keyword(s):  
Routing Algorithm ◽  
Nested Dissection ◽  
Sparse Graphs ◽  
Compact Routing ◽  
Vertex Graph ◽  
Vertex Separators ◽  
Practical Algorithm ◽  
Routing Tables ◽  
Basic Version ◽  
Advanced Version

We describe a simple and practical algorithm for compact routing on graphs which admit compact and balanced vertex separators. Using a recursive nested dissection of then-vertex graph based on these separators, we construct routing tables with as few as O(log n) entries per vertex in a preprocessing step. They support handshaking-based routing on the graph with moderate stretch, where the handshaking can be implemented similarly to a DNS lookup. We describe a basic version of the algorithm that requires modifiable headers and a more advanced version which eliminates this need and provides better stretch. A number of algorithmic parameters control a graceful tradeoff between the size of the routing tables and the stretch. Our routing algorithm is most effective on planar graphs and unit disk graphs of moderate edge/vertex density.

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