I am not sure whether this qualifies as "well-known". Anyway, in set theory, in the study of the partition calculus (transfinite generalizations of Ramsey's theorem), effort centered for a while in studying relations of the form $$ \omega^m\to(\omega^n,k)^2 $$ for $m,n,k$ positive integers. Here, exponentiation is in the ordinal sense. This re […]
This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]
One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]
Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|
In Ralph P. Boas's A primer of real functions, page 118, this is discussed in the following way: The derivative of infinite order of $f$ is defined on an interval $I$ iff the sequence $(f^{(n)})$ converges uniformly on $I$ (it is enough to require uniform convergence on compact subsets of $I$). Call $L$ the limit of this sequence, so $L$ is continuous a […]
Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alpha
The point here is that two functions are close iff they agree on an initial segment, that is, $d(f,g)\le 1/(n+1)$ iff $f(0)=g(0),f(1)=g(1),\dots,f(n-1)=g(n-1)$. Now, if $(f_n)_n$ is a Cauchy sequence, then, for each $n$, there is $N_n$ such that for all $m,k>N_n$ we have $d(f_m,f_k)\le1/(n+1)$. That is, all functions $f_m$ with $m>N_n$ agree on their f […]
No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]
This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]