CN-LEAGUE 2003 Problem 3

Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$ , $l\ge q^2(q+1)^2+1$ , $q\ge2$ , $q\in\mathbb{N}$ . Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $AB, BC, CD, DA$ in the figure.