Infinity Pizzeria offers an unlimited variety of toppings for its pizzas. One day Infinity receives an order for 1000 two-item pizzas. You notice that any three pizzas on the order have a common topping. Show that there is a topping common to all of the pizzas on the order.
Suppose the first pizza is mushroom and pepperoni. If there is no topping common to all of the pizzas then there is some pizza, say A, that doesn't have mushrooms and some pizza, say B, that doesn't have pepperoni. But then the first pizza together with pizzas A and B are three pizzas without a common topping. (If pizzas A and B were the same pizza then this pizza, the first pizza, and any other pizza would be three pizzas without a common topping.)