At least two elements have common attributes among N elements with N attributes

Question

I am writing a report, where I need to show a formula. Suppose I have n-numbers of malware samples, where each one can have attributes like 1,2,...i.

I want to say something like my theory will work only if, "at least" two malware samples have "at least" one common attribute.

I want to show something like, the theory works only if the ∩ (intersection) of "at least" two malware samples have "at least" one common attribute (is not Null).

How to formalize this in math?

Answer 1

Let M be the set of n malware samples.
Let A be the set of i attributes.
For each bad guy m in M, let a(m) be the set of attributes m has.

You requirement is:
exists distinct n,m in M with not empty a(m) $\cap$ a(n).