# Generalizing a conjecture on interval intersection

This question asks for a generalization of the conjecture posed here. It is known that if $a\in \mathbb{N}_{\geq3}$

$$\bigcap_{i = 1}^{a-1} \bigcup_{j = 0}^{i-1} \left[\frac{1}{i}\left(j+\frac{1}{a-b}\right),\frac{1}{i}\left(j+1-\frac{1}{a-b}\right)\right] = \varnothing \tag{1}$$

for all $b\geq1$.

Suppose a set $C=\{c_k\}$ such that all the elements of $C$ are distinct, $c,k\in \mathbb{N}$ and $1\leq c\leq a-1$ and $k$ runs from $1$ to $b$.

Conjecture: Forcing $i\notin{C}$ renders $(1)$ untrue.

The reference above proved the case for $b=1$ which could be visualized as removing one strip of intervals. Now the general interest is in the asking if $(1)$ is rendered false by removing $b$ strips of intervals.

-