Question

About a set X of numbers we say that it is almost sum-free if the sum of two different elements of X never belongs to X. For instance, the set {1, 2, 4} is almost sum-free. Almost-Schur number A(k) is the largest integer n for which the interval {1, . . . , n} can be partitioned into k almost sum-free sets. Use clingo to find the exact values of A(1), A(2), A(3) and try to find the largest lower bound for A(4), i.e., the largest number l such that A(4) ≥ l. Hint:​ you do not need to find all partitions to find the values of A(k).

PLEASE USE CLINGO.

Answer 1

To find the values of A(1), A(2), and A(3) for the almost sum-free sets, we can use clingo to generate the partitions of the interval. To find the largest lower bound for A(4), we can perform a similar process but stop when the computation becomes expensive.

Step-by-step explanation:

To find the exact values of A(1), A(2), and A(3), we can use clingo to generate the partitions of the interval {1, ..., n} into almost sum-free sets. We can start with n=1 and increment it until we find a value of n where the partitions cannot be formed. The largest value of n for which the partitions can be formed will be the value of A(1). Similarly, we can find the values of A(2) and A(3).

To find the largest lower bound for A(4), we can follow a similar approach but stop when the value of n is very large and generating the partitions becomes computationally expensive.