Seventeen positive integers

The following 12 numbers satisfy the conditions:

$$203,385,437,713,814,1330,1479,1495,2418,3441,11951,70499$$

Represented as a graph where the numbers are its vertices, two of which are joined by an edge if they have a common divisor greater than 1, the graph can be shown not to have a complete subgraph on four vertices ($K_4$):

nor does its complement:

In SageMath Cell Server it can be evaluated.

I must admit that I looked this up, but the largest graph which works turns out to be unique, and is called

Now all that remains is to label the vertices appropriately. One easy way to do this is to

Of course the numbers you get are rather large. You can improve this a bit as follows:

There may be a way to reduce the numbers further.