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We define the notion of -dense subsets and enumerate the number -dense subsets of the finite set .
**1. Basic**

**Definition**. A subset of is said to be -dense if such that .

**Lemma**. The number of compositions of with not more than parts and no part exceeding is defined as . By Inclusion-Exclusion principle,

**Definition**. is defined as the number of -dense subsets of with least cardinality.

We observe that this least cardinality has to be .

**2. The open problem**

The following matrix shows the values with .

\bigskip

Can we arrive at the values and the values of ‘s where the function maximizes?

(I will write more soon 🙂 )

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