Questions tagged [erdos]
For questions related to the work of Paul Erdős, especially the many results and conjectures which bear his name.
33 questions
1
vote
0
answers
147
views
Inequalities in the classic proof of perfect matching in Erdős–Rényi graph
I am checking the classic paper by Erdős and Rényi, "On the existence of a factor of degree one of a connected random graph" with the link here. I am curious about the computation of the ...
- 97
11
votes
0
answers
488
views
Are there 100 points that are part of every half-density part of the plane?
Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...
- 18.7k
8
votes
0
answers
459
views
Extension of Erdős-Gallai (s,t)-path theorem to directed graphs
The following is a result of Erdős-Gallai from 1959 (https://link.springer.com/article/10.1007/BF02024498):
Given a 2-connected undirected unweighted graph with minimum degree at least $d$, for every ...
- 81
4
votes
0
answers
195
views
Happy ending problem - why not a proof by induction? (cont)
After sharing ideas on this post, I have been thinking for some time on the problem, and I think that a possible way to prove the Erdös-Szekeres conjecture could be structured as follows:
Consider ...
- 189
17
votes
1
answer
921
views
List of problems that Erdős offered money for?
Is there a list somewhere of all the problems that Erdős offered cash awards for, including both solved and unsolved problems? One would think that the answer is yes, but so far I have had no luck ...
- 82.6k
2
votes
0
answers
123
views
Quasi-ideals and Erdős conjecture on arithmetic progressions
Starting to read a book about algebraic geometry as well as the Wikipedia article on Erdős conjecture on arithmetic progressions, I came to think of what follows.
Let $A$ be a set of positive integers,...
- 6,976
3
votes
1
answer
333
views
Probabilistic bound to the number of edge disjoint triangles in a random graph
Let $G$ be a random graph with $n$ vertices, and let $\delta(G)$ be the maximum number of triangles in $G$.
Question. How to prove the bound $$P(\delta(G)) \leq m - t \sqrt{f(m)}) \leq 2e^{-t^2 / 4}$...
user178238
5
votes
2
answers
308
views
Updates on a least prime factor conjecture by Erdos
In the 1993 article "Estimates of the Least Prime Factor of a Binomial Coefficient," Erdos et al. conjectured that
$$\operatorname{lpf} {N \choose k} \leq \max(N/k,13)$$
With finitely many ...
0
votes
1
answer
260
views
Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
- 1,826
4
votes
0
answers
441
views
The "core" of complete Erdős space
This question is about the Erdős spaces:
$\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and
$\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\}...
- 3,317
13
votes
2
answers
674
views
A reformulation of Erdős conjecture on arithmetic progressions
Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $ \forall \ell \ge 3$ the set $S$ ...
5
votes
1
answer
372
views
Countable version of Erdös-Lovasz-Faber conjecture
Let $X$ be an infinite set, and let $(A_n)_{n\in\omega}$ be a collection of subsets of $X$ with the following properties:
$|A_m\cap A_n| \leq 1$ for $m\neq n\in \omega$, and
$|A_n|=\aleph_0$ for all $...
- 48.8k
37
votes
2
answers
4k
views
How to find Erdős' treasure trove?
The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, ...
16
votes
2
answers
1k
views
Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?
As we know Erdos has proposed a considerable number of problems in the "American Mathematical Monthly" journal. Is there any published summary of Erdos's published problems in the American ...
- 551
0
votes
1
answer
1k
views
Random graphs- Erdos and Renyi 1959 paper
Please refer to this link. It is Erdos and Renyi's first paper on Random Graphs (1959). I am trying to work through it.
I'm struggling with equations (16), (17) and (21).
(16)
I'm not sure why ...
12
votes
0
answers
469
views
On a relaxed form of Goldbach's conjecture proposed by Erdős
The Goldbach's conjecture says that:
"Every even integer greater that $2$ is the sum of two prime numbers".
Let $\varphi$ denote the Euler's totient function. I remember that a long time ago I read ...
user40023
2
votes
0
answers
220
views
mixing time of random walks on dense Erdos Renyi graphs
Is there anything known about the mixing time of a simple random walk on an Erdos-Renyi graph with parameter $\langle n,d \rangle$ where $d=n^a (0<a<1 )$. I know about Reed et al and Benjamini ...
2
votes
1
answer
727
views
When an Erdos-Renyi graph is locally tree like?
I would like to know when an ER graph is locally treeing like. In this post.
I found this comment:
I think $N$ is $\log2|V|$, or something like that, in that paper.
They consider binary vectors ...
- 21
1
vote
1
answer
413
views
estimating binomial coefficients
There is a beautiful paper on the arXiv by Andrew Suk containing an asymptotic result about the Erdös-Szekeres convex polygon problem. I am struggling with one of the estimates he makes on page 4. He ...
- 7,637
18
votes
2
answers
3k
views
Many representations as a sum of three squares
Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
- 1,072
14
votes
1
answer
4k
views
What is the source of this E̶r̶d̶ő̶s̶ quote?
Namely, the following one
"All problems appeared once in the [American Mathematical] Monthly."
I remember reading it several years ago... When I first posed the question, I believed that I had ...
- 8,842
32
votes
2
answers
3k
views
The Erdős–Turán conjecture or the Erdős conjecture?
This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ and $$\...
- 11.4k
2
votes
3
answers
591
views
A limit from an Erdos paper
Hi,
I need help to prove that, for $ N = \big\lfloor \frac{1}{2}n\log(n)+cn \big\rfloor $ with $c \in \mathbb R $ and $0 \leq k \leq n: $
$$ \lim_{n\rightarrow +\infty} \dbinom{n}{k} \frac{\dbinom{\...
- 21
3
votes
0
answers
714
views
Paul Erdős and Ramanujan Primes
It's easy to find Ramanujan's proof of Ramanujan primes:
Ramanujan's Proof
Wikipedia mentions that Paul Erdős also had a proof:
Wikipedia article on Bertrand's Postulate
Does anyone know the ...
- 1,049
3
votes
1
answer
502
views
A question about the number of intersections of lines in $R^{3}$
Suppose I have n lines in $R^{3}$ with the conditions that: no 3 lines in one plane, no 3 lines intersect at one point, for fixed 2 lines, no 3 lines intersect these 2 lines at the same time.
what is ...
- 215
4
votes
1
answer
452
views
Degree reduction argument in Guth-Katz'sproof of Erdos distinct distance problem in the plane
In the middle of page 9 of
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf.
They said " Now we select a random subset....choosing lines independently with
probability $\frac{Q}{100}$. With ...
- 215
15
votes
3
answers
2k
views
Erdos distance problem n=12
The recent paper On the Erdos distinct distance problem in the plane
Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: an example of 12 ...
- 161
2
votes
2
answers
350
views
Determining the vector space for application of Cauchy Schwarz
In the paper "On the Erdős distinct distance problem in the plane" by Larry Guth and Nets Hawk Katz,
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf
they define the functions $d(P)$, (...
- 333
13
votes
2
answers
931
views
Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture?
I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the Erdős-Turán conjecture. The statement of this theorem is
Let $a_1 < ...
- 85.5k
39
votes
5
answers
3k
views
Does there exist a comprehensive compilation of Erdos's open problems?
Fan Chung and Ron Graham's book Erdos on Graphs: His Legacy of Unsolved Problems (A. K. Peters, 1998) collects together all of Erdos's open problems in graph theory that they could find into a single ...
- 82.6k
3
votes
1
answer
859
views
A limit involving the totient function
P. Erdős and Leon Alaoglu proved in [1] that for every $\epsilon > 0$ the inequality $\phi(\sigma(n)) < \epsilon \cdot n$ holds for every $n \in \mathbb{N}$, except for a set of density $0$.
C. ...
- 8,842
15
votes
6
answers
5k
views
If Erdős is published as Erdös in a paper, which do I cite?
There seems to be a few papers around with Erdős written as Erdös. For example:
MR0987571 (90h:11090) Alladi, K.; Erdös, P.; Vaaler, J. D. Multiplicative functions and small divisors. II. J. Number ...
28
votes
5
answers
9k
views
Erdos Conjecture on arithmetic progressions
Introduction:
Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length.
Question:
I ...
- 4,952