When neutral charge comes into contact with a positive charge

If they come in direct contact, then with certain restrictions, the neutral charge will experience a transfer of 1/2 a charge from the positive body providing we're talking about a large enough charge that can be transferred.

So if one sphere starts with no charge and one with 6 * 10^-14 Q, if they touch, then each sphere will have a charge of 3*10^-14 Q which is still a pretty big charge.