$\newcommand\R{\mathbb R}$Let $0<d\_1<\cdots<d\_k<\infty$ and let $m\_1,\dots,m\_k$ be any integers $\ge1$. Let $n:=m\_1+\dots+m\_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.



> 
> Do then there always exist pairwise disjoint subsets $S\_1,\dots,S\_k$ of $\R^n$ of respective cardinalities $m\_1,\dots,m\_k$ such that
> $$d(x,y)=d\_j$$
> for all $j\in\{1,\dots,k\}$ and all $(x,y)\in T\_j\times S\_j$ with $x\ne y$, where $T\_j:=S\_1\cup\dots\cup S\_j$?
> 
> 
> 


This conjecture is obviously true for $k=1$ and seems not hard to prove for $k=2$.




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Reviewing the edits suggested by MattF., one can restate the conjecture in the following way:



> 
> Let $n$ be any natural number. Suppose that $0<d\_1\le\cdots\le d\_{n+1}<\infty$.
> 
> 
> 



> 
> Do then there always exist pairwise distinct points $x\_1,\dots,x\_{n+1}$ in $\R^n$ such that
> $$d(x\_i,x\_j)=d\_j$$
> for all integers $i,j$ such that $1\le i<j\le n+1$?
> 
> 
> 



About Euclidean distances
Finite [ultrametric spaces](https://en.wikipedia.org/wiki/Ultrametric_space) [allow isometric embeddings into $L\_2$](https://core.ac.uk/download/pdf/82490074.pdf).


