Ideal of $\mathbb{C}[x,y]$ not generated by two elements
Set $I=\langle xy^3, x^2y^2, x^3y\rangle$ and $\mathfrak m=\langle x, y\rangle$.
$\dim I/\mathfrak mI=3$
Just show that the residue classes of the generators of $I$ are linearly independent over $K$, that is, if $axy^3+bx^2y^2+cxy^3\in mI$ with $a,b,c\in K$ then $a=b=c=0$. This is obvious since the generators of $\mathfrak mI$ are homogeneous of degree $5$.
If $\dim I/\mathfrak mI=3$ then the minimal number of generators of $I$ is $3$.
This is very well explained here.