scholarly journals A short proof of van der Waerden’s theorem on arithmetic progressions

1974 ◽  
Vol 42 (2) ◽  
pp. 385-385
Author(s):  
R. L. Graham ◽  
B. L. Rothschild
Keyword(s):  
Short Proof ◽  
Arithmetic Progressions

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 Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

2013 ◽  
Vol 22 (3) ◽  
pp. 351-365 ◽  
Author(s):  
ERNIE CROOT ◽  
IZABELLA ŁABA ◽  
OLOF SISASK
Keyword(s):  
Arithmetic Progression ◽  
Short Proof ◽  
Arithmetic Progressions ◽  
Almost Periodicity ◽  
General Lemma ◽  
Formula Group ◽  
Alpha Beta ◽  
Image Position ◽  
Sum Of Functions

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation} Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation}

scholarly journals On Arithmetic Progressions of Cycle Lengths in Graphs

2000 ◽  
Vol 9 (4) ◽  
pp. 369-373 ◽  
Author(s):  
JACQUES VERSTRAËTE
Keyword(s):  
Bipartite Graph ◽  
Short Proof ◽  
Minimum Degree ◽  
Average Degree ◽  
Arithmetic Progressions ◽  
Cycle Lengths

A question recently posed by Häggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

A note on generalized quasi-contraction

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3091-3093
Author(s):  
Dejan Ilic ◽  
Darko Kocev
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Short Proof

In this paper we give a short proof of the main results of Kumam, Dung and Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015), 1549-1556).

Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

2021 ◽  
Vol 344 (7) ◽  
pp. 112430
Author(s):  
Johann Bellmann ◽  
Bjarne Schülke
Keyword(s):  
Short Proof ◽  
Kneser Graphs

scholarly journals Strongly perfect claw‐free graphs—A short proof

2021 ◽  
Author(s):  
Maria Chudnovsky ◽  
Cemil Dibek
Keyword(s):  
Short Proof

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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