Question: I would like to organize a game playing session for 140 people in the same room. I could just split them up into lots of small groups and organize a tournament. But then I started to wonder, are there any massively multiplayer non-online games? My restriction is that any material needed will have to be printed out.

Answer: 
Once the second Pledge enters the battlefield, the first one again starts granting protection from white to the Gnomes, which causes the second aura to fall off. This happens as a state-based action just after the second Pledge resolves.
See Kevin's answer: the second Pledge causes the first to also grant protection from white.

If somehow both Pledges are colorless, and then both stop being colorless at end of turn, then my original answer applies:

Both Pledges will go to the graveyard, for exactly the reason you state. As soon as the Gnome's effect ends, state-based effects are checked and both auras are enchanting an illegal permanent. The Gnomes have two "instances" of protection from white, each of which doesn't remove one copy of Pledge of Loyalty, but not the other. Since all state-based effects are checked at once, both Pledges go to the graveyard simultaneously.