On Lower Order Extremal Integral Sets Avoiding Prime Pairwise Sums

Let A be a subset of {1, 2, . . . , n} such that the sum of no two distinct elements of A is a prime number. Such a subset is called a prime-sumset-free subset of {1, 2, . . . , n}. A prime-sumset-free subset is called an extremal prime-sumset-free subset of {1, 2, . . . , n} if A ∪ {a} is not a prime-sumset-free subset for any a ∈ {1, 2, . . . , n} \ A. We prove that if n ≥ 10 then there is no any extremal prime-sumset-free subset of {1, 2, . . . , n} of order 2 and if n ≥ 13 then there is no any extremal prime-sumset-free subset of {1, 2, . . . , n} of order 3. Moreover, we prove that for each integer k ≥ 2, there exists a nk such that if n ≥ nk then there does not exist any extremal prime-sumset-free subset of {1, 2, . . . , n} of length k. Furthermore, for some small values of n, we give the orders of all extremal prime-sumset-free subset of {1, 2, . . . , n} along with an example of each order, and we give all extremal prime-sumset-free subsets of {1, 2, . . . , n} of orders 2 and 3 for n ≤ 13.