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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 $P$ …

Also not an answer, but may be useful. A somewhat similar kind of word is mentioned at the end of section 1.5 of Salomaa: Jewels of Formal Language Theory: There is an infinite word over a 3-letter …

A word is cubefree if it cannot be written as $xyyyz$ where $y$ has positive length. Let $h$ be the morphism from $\{0,1,2,3,4\}^*$ to $\{0,1\}^*$ given for words of length 1 as follows ($a\to h(a)$) …

Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is combinatorics but similar things come up sometimes in theoretical computer science, for examp …

Yes, this should follow from the elementary bound. The point is that having a Kalmar elementary time bound is "closed under" searches through exponentially large collections. Suppose $N=W(r,k)$ is lea …

According to the 2013 paper "Computing the Partial Word Avoidability Indices of Ternary Patterns" by Blanchet-Sadri, Lohr, and Scott, The problem of deciding whether a given pattern is avoidable h …

V. Dotsenko's construction, on math.stackexchange: https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450 may fit your requirement "combinatorial mapping …

Let us choose $\alpha$ many sets of size $n/3$ at random. The cardinality $X=|U\cap V|$ of the intersection of two of them is a hypergeometric random variable (well, given $U$, but the value of $U$ do …

Regarding the 3rd question, I will show this: Theorem. For a random binary word of length $n$, the expected number of $h$th powers is $$ \sim \frac{n}{2^{h-1}-1}. $$ Proof. A basic event about occurr …

The problem to determine whether two 4-manifolds, given as simplicial complexes, are homeomorphic. This was shown to be undecidable by Markov. (Some theories of physics involve a sum over such manifol …

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. A n …

Yes, this has been studied and is indeed known as ordered geometry or the study of betweenness spaces: https://en.m.wikipedia.org/wiki/Ordered_geometry

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

Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the compleme …

This is $\mathbb E (X_1^{n-1} X_2) $ where $ X_i $ is the $ i $ th order statistic of a sample from the uniform distribution on $[0, t] $. To evaluate it can try using the joint pdf of these order sta …