If $X$ is a set and $A,B\subseteq X$ we let the Hamming distance of $A,B$ be defined as $d_H(A,B) = \big|(A\setminus B)\cup(B\setminus A)\big|$. If $\newcommand{\S}{{\cal S}}\S\subseteq {\cal P}(X)$, let the discrepancy of ${\cal S}$ be definied by $\newcommand{\d}{\text{discr}}\d(\S)=\min\{d_H(A,B): A\neq B\in\S\}$.
For $n\in\mathbb{N}$, let $[n]=\{1,\ldots,n\}$. Let $X$ be a set and $n\in\mathbb{N}$. Then $[X]^n$ denotes the collection of subsets of $X$ having $n$ elements.
For $n\in\mathbb{N}$, let $M_n:= \max\{\d(\S):\S\in \big[{\cal P}([n])\big]^n\}$.
Question. Is there an explicit formula for Experiments by $M_n$ for allRob Pratt and $n\in\mathbb{N}$?
EDIT. I proposeClaude Chaunier as hypotheseslaid out in the comments below (cheers to Rob and Claude!) seem to point to the following "almost explicit formulas"hypothesis:
- $M_{2k} = k$ , and
- $M_{2k+1} \in \{k, k+1\}$
For all $k\in\mathbb{N}$ with $k\geq 2$ we have
- $M_{2k} = k$ , and
- $M_{2k+1} = k$ for $k$ even, and $M_{2k+1} = k+1$ for $k$ odd.
for all $k\in\mathbb{N}$ with $k\geq 2$.Question. Is the above hypothesis true?
