I'm at work on mobile and can't type lengthy replies. I already told them to read the Constitution and Washington's final address.
The insistence that the Consitution has anything to do with the two-party system is just wrong. We had an 'Era of Good Feelings' beginning in 1816 under Monroe whose ultimate goal was to eliminate parties altogether.
The two-party system emerged from a variety of factors. Blaming it on the Constitution is extremely limited.
Edit: Sorry I was harsh everyone. I shouldn't have been so pejorative. Happy to continue this discussion at a later point with anyone once I'm at a place where I'm not typing on an iPhone 5 with a broken screen >.<
Washington's farewell address is not law, if it was then the country would be far more isolationist than it is.
Until the 12th Amendment gets changed, the two party system is absolutely entrenched. You simply can not have a 3 or 4 party system where it is a requirement for the victor to win a clear majority. You simply can't.
Your problem seems to stem from the common mistake of treating the Founding Father as if they were a unified group. There were tons of disagreements. Just because Washington and Monroe wanted something doesn't mean that everyone wanted it or that it was a good idea. They spoke out against the idea of political parties and they lost out.
Intentional or not, the two party system is written into the Constitution. How would you propose we go about creating a multiparty system under the current constitution? What would that even look like? Nobody would ever make it to 270. Every election would come down to the current Speaker of the House's choice. You would effectively remove a branch of government. The White House would just become an arm of the Legislative branch.