Any set S of integers which contains [abcd] = 1,4,2,3 will meet the condition that there are distinct pairs which are equal (i.e. 1+4 = 2+3). The minimal EN would have to be 20, since S must have exactly 20 distinct members (i.e. 1..20). There is no "maximal value of EN" if the members are not constrained to exclude the subset [1,2,3,4].

I assume the puzzle must mean something like: what is the smallest limit (for a set of twenty members each distinct and lower than EN) for which the sum of no two equals the sum of any other two members. This would then give limit-1 as the maximal value for which the claim of always at least one sum would hold. One would need to show how the 20 numbers are chosen to preclude any pair sums (or produce such a set). If a pseudo-Fibonacci fomula were used e.g. starting with 1,2 then each number equalled the sum of the previous two + 1), then there would be no same sums for any two pairs of members (20th term 17710). I'm sure someone will find a clearer way to represent the task.