The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner that will be made precise below.

## Setup

Consider a nested family of simplicial complexes $K(p)$ indexed by prime $p \in \mathbb N$ defined as follows:

- the vertices are all primes less than or equal to $p$, and
- insert a $d$-simplex ($d \geq 2$) spanning $d+1$ vertices if and only if they constitute an arithmetic progression. Of course, one must also insert all faces, and faces-of-faces etc. so that the defining property of a simplicial complex is preserved.

For instance, $K(7)$ has the vertices $2,3,5,7$ and a single $2$-simplex $(3,5,7)$ along with all its faces. $K(11)$ has all this, plus the vertex $11$ and the simplex $(3,7,11)$. The edge $(3,7)$ already exists so only the other two need to be added. Thus, the fact that $(3,7)$ occurs in two arithmetic progressions bounded by $11$ is encoded by placing the corresponding edge in the boundary of two simplices.

## Question

Has someone already defined and studied this complex? What I am mostly interested in is

How does the homology of $K(p)$ change with $p$?

It is easy to check that the only interesting homology is in dimensions $ \leq 1$. If it helps, here are -- according to home-brew computations -- the statistics for the first few primes (Betti 0 and 1 over $\mathbb{Z}_2$). I've already confirmed that the sequence of Betti-1's is not in Sloane's online encyclopedia of integer sequences. If an intermediate K[p] is missing in the list, that means that the homology is the same as that for the previous prime.

## Update 1##

Zack pointed out an error in the previous computations, so here are the betti numbers with that error fixed. I have also removed "2" from the vertex set since the only contribution of that vertex to the homology is incrementing all the $0$ dimensional betti numbers by +1.

K [3]: 1 0

K [5]: 2 0

K [7]: 1 0

K [13]: 2 0

K [17]: 2 1

K [19]: 1 2

K [23]: 1 4

K [31]: 1 6

K [37]: 2 6

K [43]: 1 7

K [53]: 1 8

K [59]: 1 9

K [61]: 1 10

K [67]: 1 12

K [71]: 1 17

K [73]: 1 20

K [79]: 1 23

K [83]: 1 26

K [89]: 1 31

K [97]: 1 32

K [101]: 1 35

K [103]: 1 41

K [107]: 1 43

K [109]: 1 47

K [113]: 1 53

K [127]: 1 58

K [131]: 1 62

K [137]: 1 67

K [139]: 1 73

K [149]: 1 78

## Update 2

I've just finished running the homology computations (over $\mathbb{Z_2}$) for all primes less than $30,000$, and this text file contains the resulting Betti numbers. Once such data is available for control experiments, such as Cramer numbers or primes which are $1$ mod 4, I will put up those text files as well. Also, I am no longer confident that higher homology will not appear, so here is an auxiliary question:

Is it true that the higher homology groups of $K(p)$ are trivial?