1 Basic Sidon sets results

Definition 1 Sidon set

✓

A set $S \subseteq M$ , where $M$ is an additive commutative monoid, is called a Sidon set if for all $a, b, c, d \in S$ , then $a + b = c + d ⟹ {a, b} = {c, d}$ , that is, there are no repeated sums.

Theorem 2

✓

The empty set is a Sidon set.

Theorem 3

✓

A singleton set is a Sidon set.

Theorem 4

✓

If $S$ is a Sidon set and $T \subseteq S$ , then $T$ is a Sidon set.

Theorem 5

If $S$ is a Sidon set in a finite additive commutative group $M$ , then $| S | \cdot (| S | - 1) \leq | M |$ .

Proof ▶

Let $S = {a_{1}, \dots, a_{n}}$ . Consider the differences $a_{i} - a_{j}$ for $i \neq j$ . Since $S$ is Sidon, theses differences are unique, and there are at most $| M |$ of them. By injectivity, the proof follows. □