Satisets

Satisets are a conceptual mathematical object defined as a finite set of positive integers such that for any two distinct elements a and b in the set, their sum a+b is not present in the set. This condition is crucial to the definition. For example, the set {1, 2, 4} is a satiset because 1+2=3 (not in the set), 1+4=5 (not in the set), and 2+4=6 (not in the set). The set {1, 2, 3} is not a satiset because 1+2=3, and 3 is an element of the set.

The study of satisets falls within the broader field of additive combinatorics, which investigates the additive properties of sets of integers. Researchers are interested in questions such as the maximum size of a satiset within a given range of integers, or the existence of satisets with specific properties. For instance, a satiset cannot contain the number 1, because if it did, then 1+1=2 would have to be in the set, and if 2 were in the set, then 1+2=3 would have to be in the set, and so on, potentially leading to an infinite set or a contradiction if the set is finite and the sums remain within the set.

The concept of a satiset is related to other combinatorial structures, such as Sidon sets or $B_2$ sets, which have slightly different definitions but share similar themes of avoiding sums within the set or relating to sums of pairs. The exploration of satisets contributes to understanding the combinatorial constraints imposed by additive relationships within finite sets of numbers.