(In reply to re(2): No Subject
We are seeking the SMALLEST value of EN for which no set S of 20 members -- each distinct, > 0, and < EN -- lacks the property that at least four members are related such that a+b = c+d. "no set lacks" = "every set satisfies"erNoNo
Your original posting showed a method to prove easily that for certain values of EN the addition test is always satisfied. If there are 20 members in S, then there are n*(n-1)/2 = 190 sums the largest of which could be (EN-1 + EN-2) and the smallest of which could be 3 (i.e. 1+2). If EN were 21, there would be 190 sums, but all in the range 3..39 -- obviously forcing many duplicate sums. If EN = 98, then the 190 sums could be in the range 3..193, so duplication might be avoidable -- but if EN=97 then the range shrinks to 3..191, so some duplication would always occur.
Does this prove that EN=97? It only shows the lower limit for a possible largest EN. But some other analysis would have to defend a larger value -- and exhaustive enumeration of all 20-subsets of a larger EN would quickly become a computational monster. If we tried to introduce the Fibonacci element, we are avoiding the requirement that S may be any subset of the numbers from 1 to EN-1. Now what??