open intervals intersection

Question

Here I'm seeking a reference of any sort about this question. It is known that two open intervals $(a_1,b_1)$ and $(a_2,b_2)$, intersect if and only if $$a_1 < b_2$$ and $$b_1<a_2.$$ Now, I have 4 open intervals $I_1=(a_1,b_1)$,$I_2=(a_2,b_2)$,$I_3=(a_3,b_3)$, and $I_4=(a_4,b_4)$, where $a_i>0$, $b_i>0$ for $i=1,2,3,4$. Suppose that these intervals intersect, is there any necessary and sufficient conditions similar to the case above? are there any references talking about such topics? any idea is appreciated!!