Szemerédi's regularity lemma application on 3-term arithmetic progression

2020 ◽  
Author(s):  
Regina Ayunita Tarigan ◽  
Chun-Yen Shen
Keyword(s):  
Arithmetic Progression ◽  
Regularity Lemma ◽  
Szemeredi's Regularity Lemma

scholarly journals Tilings in Randomly Perturbed Dense Graphs

2018 ◽  
Vol 28 (2) ◽  
pp. 159-176 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
ANDREW TREGLOWN ◽  
ADAM ZSOLT WAGNER
Keyword(s):  
High Probability ◽  
Minimum Degree ◽  
Graph Model ◽  
Regularity Lemma ◽  
Arbitrary Graph ◽  
Random Graph Model ◽  
Dense Graphs ◽  
Szemeredi's Regularity Lemma ◽  
Vertex Disjoint ◽  
Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

scholarly journals Stability for Vertex Cycle Covers

10.37236/5185 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
József Balogh ◽  
Frank Mousset ◽  
Jozef Skokan
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Stability Result ◽  
Natural Generalization ◽  
Regularity Lemma ◽  
Property A ◽  
Vertex Graph ◽  
Szemeredi's Regularity Lemma ◽  
Cycle Covers ◽  
Positive Integers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.

scholarly journals Some Properties of Lower Level-Sets of Convolutions

2012 ◽  
Vol 21 (4) ◽  
pp. 515-530
Author(s):  
ERNIE CROOT
Keyword(s):  
Level Set ◽  
Arithmetic Progression ◽  
Level Sets ◽  
Regularity Lemma

In the present paper we prove a certain lemma about the structure of ‘lower level-sets of convolutions’, which are sets of the form {x ∈ ℤN : 1A * 1A(x) ≤ γ N} or of the form {x ∈ ℤN : 1A(x) < γ N}, where A is a subset of ℤN. One result we prove using this lemma is that if |A| = θ N and |A+A| ≤ (1 − ϵ)N, 0 < ϵ < 1, then this level-set contains an arithmetic progression of length at least Nc, c = c(θ, ϵ, γ) > 0. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with some ideas of Bourgain [6]); however, our method of proof allows us to obtain non-tower-type quantitative dependence between the constant c and the parameters θ and ϵ. For various reasons (discussed in the paper) one might think, wrongly, that such results would only be possible for level-sets involving triple and higher convolutions.

scholarly journals On the Distribution of Three-Term Arithmetic Progressions in Sparse Subsets of Fpn

2011 ◽  
Vol 20 (5) ◽  
pp. 777-791
Author(s):  
HOI H. NGUYEN
Keyword(s):  
Arithmetic Progression ◽  
Short Proof ◽  
Arithmetic Progressions ◽  
Regularity Lemma ◽  
Random Set ◽  
Almost All

We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fpn. For every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cpn/2 and for almost all sets R of size r of Fpn. Let A be any subset of R of size at least αr; then A contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, Łuczak and Rödl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right.

scholarly journals Szemerédi's Regularity Lemma via Martingales

10.37236/5585 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Pandelis Dodos ◽  
Vassilis Kanellopoulos ◽  
Thodoris Karageorgos
Keyword(s):  
Random Variables ◽  
Regularity Lemma ◽  
Martingale Difference ◽  
Difference Sequences ◽  
Szemeredi's Regularity Lemma ◽  
Probabilistic Version

We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.

scholarly journals A Simple Algorithm for Constructing Szemerédi's Regularity Partition

10.37236/1449 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Alan Frieze ◽  
Ravi Kannan
Keyword(s):  
Simple Algorithm ◽  
Singular Values ◽  
Regularity Lemma ◽  
Constructive Version ◽  
Szemeredi's Regularity Lemma

We give a simple constructive version of Szemerédi's Regularity Lemma, based on the computation of singular values of matrices.

scholarly journals On the Size-Ramsey Number of Cycles

2019 ◽  
Vol 28 (06) ◽  
pp. 871-880
Author(s):  
R. Javadi ◽  
F. Khoeini ◽  
G. R. Omidi ◽  
A. Pokrovskiy
Keyword(s):  
Upper Bounds ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Szemeredi's Regularity Lemma ◽  
Number Of Cycles ◽  
Image Position ◽  
Edge Colouring ◽  
Monochromatic Copy

AbstractFor given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

Sign in / Sign up

Export Citation Format

Share Document