Pigeonhole principle

The pigeonhole principle is the idea that if $𝑛$ items are put into $𝑚$ containers with $𝑛 > 𝑚$ , then at least one of said containers must contain multiple items.

A stronger version states that given $𝑚, 𝑘 \in ℕ 0$ , and set $𝐼$ of $𝑛 : = 𝑘 𝑚 + 1$ items partitioned into sets ${𝑆 𝑖} 𝑛 𝑖 = 1$ , then $| 𝑆 𝑖 | \geq 𝑘 + 1$ for some $1 \leq 𝑖 \leq 𝑛$ .