Questions tagged [erdos]
For questions related to the work of Paul Erdős, especially the many results and conjectures which bear his name.
39 questions
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Planted Matching in Hypergraphs
Cross posted from MSE
I’m working on the planted perfect matching problem in random $k$-uniform hypergraphs $k \ge 3$, and I’m stuck on rigorizing the impossibility (lower bound) side of what looks ...
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No real roots of $\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}}$
Prove or disprove that $$\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}} \neq 0$$ for any real number $t$.
I find it surprising that so simple looking equations involving complex numbers ...
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Is the least common multiple sequence $\text{lcm}(1, 2, \dots, n)$ a subset of the highly abundant numbers?
I've been comparing the sequence of the Least Common Multiple of the first $n$ integers, $L_n = \text{lcm}(1, 2, \dots, n)$, with the sequence of Highly Abundant Numbers (HA).
The two sequences in ...
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Divisibility property of colossally abundant numbers
The sequence of colossally abundant (CA) numbers, $a(n)$ (OEIS A004490), consists of positive integers that maximize the ratio $\frac{\sigma(m)}{m^{1+\epsilon}}$ for some $\epsilon > 0$.
A known ...
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Proof of every Highly Abundant Number greater than 3 is Even
A natural number $n$ is defined as a Highly Abundant Number ($\text{HAN}$) if and only if the sum of its divisors $\sigma(n)$ is strictly greater than the sum of the divisors of any natural number $m$ ...
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Bounds for number of (Erdős–Rado) sunflowers of a family of k-sets?
The Erdős–Rado sunflower conjecture asks for the minimum size $\operatorname{Sun}(k,r)\in \mathbb N$ at which families $\mathscr F$ of $k$-sets must contain an $r$-sunflower.
A common technique of ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 $$P(\delta(G)) \leq m - t \sqrt{f(m)}) \leq 2e^{-t^2 / 4}$...
user178238
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Updates on a least prime factor conjecture by Erdos
In the 1993 article "Estimates of the Least Prime Factor of a Binomial Coefficient," Erdos et al. conjectured that
$$\operatorname{lpf} {N \choose k} \leq \max(N/k,13)$$
With finitely many ...
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Generalized Erdős multiplication table problem
Consider multiplication operation $$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$ ...
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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$}\}...
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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$ ...
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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 $...
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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, ...
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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 ...
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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 they ...
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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
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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 ...
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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 of ...
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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 ...
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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 ...
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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 read ...
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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 $$\...
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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{\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $d(P)$, (...
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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 < ...
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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 ...
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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. ...
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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 ...
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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 ...
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