There are two main facts to understand about the frequency with which arithmetic progressions appear among the whole numbers.
In 1975, Peluse’s formula is complicated to state. It involves taking a double logarithm of the length of the initial interval from which you choose the numbers in your set. The minimum size she came up with is also not necessarily the true minimum size — future work might find that the critical threshold is even lower. But before her proof, mathematicians had no quantitative understanding at all about when polynomial progressions were guaranteed.