Just for the record, here are links to the 30 questions currently (Oct 18, 2022 18:20 GMT) tagged erdos. Indeed, in case the tag is eliminated, this list might fail to be produceable. The following 6 have no primary tag: Happy ending problem - why not a proof by induction? (cont), When an Erdos-...

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 was wondering how one might go about categorizing or generating the divergent series of the form i...

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 Theory 31 (1989), no. 2, 183--190. (Reviewer: Friedrich Roesler) 11N37 Would it be incorrect ...

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. L. mentioned in [2] that as a consequence of the previous result one can ascertain that $\displaystyle...

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 volume, complete with bounties where applicable. Of course Erdos posed many other open problems ...

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 $a_1 < a_2 < \cdots$ and denote by $g(n)$ the number of solutions to $n=a_ia_j$. Then $g(n)>0$ for all $n>...

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 $a_1 < a_2 < \cdots$ and denote by $g(n)$ the number of solutions to $n=a_ia_j$. Then $g(n)>0$ for all $n>...

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 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has...

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 positive probability.... I can not see why there is positive probability... Could any one expl...

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 the up bound of the number of intersections? The up bound $n^{\frac{3}{2}}$ is a simple corollar...

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 full citation for Erdős's proof that for any number $n$, there exists a prime $p$ such that for all...

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: $ $$ \lim_{n\rightarrow +\infty} \dbinom{n}{k} \frac{\dbinom{\binom{n - k}{2}}{N} }{\dbinom{\binom{n}{2} }{N}} = \frac{e^{-2kc}}{k!} $$

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 $$\sum_{a\in A} \frac{1}{a}$$ diverges, then $A$ contains arbitrarily long arithmetic progressi...

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 it somewhere in Krantz' Mathematical Apocrypha but according to Carlo Beenakker the quote is ...

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 have as a sum of three squares. Gauss proved that $r_3(n) = \frac{A\sqrt{n}}{\pi}\sum_{m-1}^\inf...

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 claims that for $n$ large enough, we have $$ 2^{n^{4/5} + 4n^{3/5} - 50n^{3/10}} \geq \binom{n + ...

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 length $N$. Furthermore, "most" sparse graphs have logarithmic diameter (say, random regular...

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 et al's result when $d=\log^a (n)$. But I need to have a denser random graph.

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 that Erdős, motivated by the identity $\varphi(p) = p - 1$ for all prime numbers $p$, asked if th...

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 are using those two bounds in the summation? It seems redundant Why they did not use the resul...

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 Mathematical Monthly journal? Thank you!

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, approximation theory, set theory, and probability theory. A complete list of his published works is av...

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 $n\in \omega$. We consider the following statement: (EFL$_\omega$:) There is $f:X\to \omeg...

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 $S$ contains an arithmetic progression of length $\ell$. In other words, this conjecture states that f...

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 $n<\omega$}\},$ where $\ell^2$ is the Hilbert space, $\mathbb Q$ is the set of rational numbers, and $\mathbb P=\m...

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 $k =2$ with $n_1=n_2$ this is the standard Erdos multiplication table problem whose estimates are in Disti...

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 exceptional $(N,k)$. Here, $\operatorname{lpf}(x)$ denotes the smallest prime factor of $x$. I am posting he...

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 $$P(\delta(G)) \leq m - t \sqrt{f(m)}) \leq 2e^{-t^2 / 4}$$ using Talagrand's concentration Inequality? Here $m$ denotes the median of $\delta(G)$. I am ...

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, and define a $k$-quasi-ideal $J$ of $A$ as a set made of $k+1$ consecutive integers such that t...

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 finding such a list. Greg Kuperberg compiled a partial list, which he posted in the answer to anothe...

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 some set $X$ of $2^{n-2}$ points in general position in the plane. We can introduce a coordinate sys...

I don't know if you have seen this but there are papers devoted to "discrete fractional calculus". Like this one for example http://arxiv.org/abs/0911.3370 or http://www.math.u-szeged.hu/ejqtde/sped1/103.pdf . Like in fractional calculus, of course the discrete fractional integral is easier to de...

Is there an account in English of results from "Homologie nicht-additiver Funktoren. Anwendungen" by Dold and Puppe? I am mostly interested in the spectral sequence of cross-effects which computes the homology of a functor on the suspension (p.251) and in the fact that higher symmetric powers of ...

Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?