1. There is a set of basketball uniform next to the basketball.

2. There is basketball uniform next to the basketball.

3. There are basketball uniforms next to the basketball.

4. There is a set of basketball uniform next to the basketball.

(When there is a uniform, both the shirt and the bottom (a short), which expression do we have to use?)

1, 2, and 4 are incorrect.

3 is fine.

A uniform is the set -- the shirt and the shorts.

The very tall player's uniform is on the bench next to the basketball. (one player's uniform)

There are several uniforms on the bench next to the basketball. (more than 3 players' uniforms)

When you have a complete set of basketball uniforms, including both the shirt and the bottom (shorts), you would use the expression "a set of basketball uniforms."

So, in the given statements:
1. "There is a set of basketball uniform next to the basketball."
2. "There is basketball uniform next to the basketball."
3. "There are basketball uniforms next to the basketball."
4. "There is a set of basketball uniform next to the basketball."

Sentence 1 is incorrect because "uniform" should be plural to match the word "uniforms" in "a set of basketball uniforms."
Sentence 2 is incorrect because it is missing the plural form of "uniforms" to indicate that there is more than one basketball uniform.
Sentence 3 is correct because it uses the plural form "basketball uniforms."
Sentence 4 is incorrect because it uses the singular form "uniform" instead of the plural form "uniforms."

So, the correct expression is "There are basketball uniforms next to the basketball."