Are there arbitrarily large sets $\mathcal S=\{a_1,\ldots,a_n\}$ of strictly positive integers such that all sums $a_i+a_j$ of two distinct elements in $\mathcal S$ are squares?

Considering subsets in $\mathbb Z$ should essentially give the same answer since such a set can contain at most one negative integer.

An example of size $3$ is given by $\{6,19,30\}$. (Allowing $0$, one gets $\{0,a^2,b^2\}$ in bijection with Pythagorean triplets $c^2=a^2+b^2$.)

There is no such example with four integers in $\{1,\ldots,1000\}$. (Accepting $0$, solutions are given by Euler bricks: $\{0,44^2,117^2,240^2\}$ is the smallest example. I suspect thus that there are strictly positive solutions in $\mathbb N^4$).

An equivalent reformulation: Consider the infinite graph with vertices $1,2,3,\ldots$ and edges $\{i,j\}$ if $i+j$ is a square. Does this graph contain arbitrarily large complete subgraphs? (Trivial observation: Every edge $\{a,b\}$ is only contained in finitely many different complete subgraphs.)

*Motivation* This is somehow a variation on question Generalisation of this circular arrangement of numbers from $1$ to $32$ with two adjacent numbers being perfect squares