How do we know that the prime sum graph defined in this paper has a perfect matching?

Question

https://arxiv.org/pdf/1804.07104.pdf

I am reading this paper and trying to understand why Theorem 1.1 guarantees a perfect matching for the prime sum graph as stated at the bottom of page 2. Can anyone provide some insight?

Thanks so much!

Answer 1

Let $q$ be a prime satisfying $2n < q <4n$. There always exists such a prime, by Chebychev's Thm proved in 1852 (formerly known as Bertrand's postulate). Then for each $i =0,1,2,\ldots q-2n$, match $2n-i$ with $q-2n+i$. Then note that $q-2n$ is an odd integer strictly less than $2n$ and that every integer in $[q-2n,q-2n+1,\ldots,2n-1, 2n]$ is matched. Thus, for some even integer $m<n$, the set of integers not yet matched are precisely those in the set $\{1,\ldots, 2m\}$. Can you finish from here?

[For $n=1$ note that $1+2=3$ a prime.]