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Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I …

Let's say that a function $d:S\times S\to [0,\infty)$ for a countable set $S$ is a metric in the limit if $$d(x,y)\le \liminf_{n\to\infty} d(x,z_n)+d(z_n,y),$$ $$\lim_{n\to\infty} d(z_n,z_n)=0, \quad\ …

Here is a counterexample of size 23. Let $m=6$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i<j\le m\}$$ where $0<a_i<b_{jk}<1$ whenever $i$ is distinct from $j$ and $k$. The cardinality of $P$ …

Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here. For instance if $W_t$ is 1-dimensional standard Brownian motion at time $t$ and $$P(\forall x\,F(x)=x^2)=1$$ and $Y=W …

Imagine that you lay out the N (0) and E (1) moves as follows ($n=4$ shown): $$0000$$ $$1111$$ As you go along the path, color $\color{red}{red}$ the ones you have used, so that after reading either 0 …

This is studied in Reverse Mathematics as the Chain Antichain Principle (CAC) and it is observed that it follows from Ramsey's theorem.

$\mathscr F$-indistinguishability. In analogy with Topological indistinguishability.

Question 1: Inspired by the ones you found we can see that there are infinitely many solutions as follows: $$(x,y,z;n) = (k-1,\quad k(k-1),\quad k-1;\quad 2^k)$$ for any $k\ge 0$. Edit re: Question 2 …

Chebyshev's bias says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).

Maybe the simplest counterexample? Let $\newcommand{\1}{\mathbf 1}\1=1_{[0,1]}$. Then any $f\cdot\1$ is zero outside of $[0,1]$, but $$\1*\1(x)=\int \1(t)\1(x-t)\,dt = \begin{cases}x& 0\le x\le 1\\ 2 …

It seems related to optimal stopping. First you have to decide when to make the first purchase. Then you have a new problem of when to make the next purchase. Although it's more complicated than that …

It seems totally fine. Maybe instead of "complete over $C'$" use "complete for $C'$" or "complete invariant of path graphs".

A year after you posted the question, Fritz showed the common theory of all such lattices is undecidable: https://arxiv.org/abs/1607.05870 In reponse to @MattF's query I'll post an example of how i …

The example you gave extends as follows: SAT for arbitrary lattices (meaning, is a given formula satisfiable in some lattice) is polynomial-time decidable SAT for modular lattices is Turing undecida …

I don't know that your characteristic has been explicitly studied before, nor would I be surprised if it has, but it fits into a more general setting as follows. The directed graph distance $d(a,b)$ …