I was in Austin Texas visiting my nephew Jason. I was also giving a talk at the University. My nephew fixes forklifts for a living and does not care about abstract things. (I'd be as bad at his job as he would be at mine.) He asked me what the talk was on. I think he asked just to be polite but did not really care. Even so, I tried to put it in as concrete terms as possible. My thought process: (1) Don't use n. Use an actual number, say 100. (2) Don't state a theorem. State an easily grasped corollary.
BILL: Picture that there is a 100 by 100 chessboard. Picture that you want to put queens on the diagonal so that every square of the board is either covered or under attack. What is the least number of queens you need for this? (NOTE: I leave it to my readers to prove that, for an n x n chessboard, this is IDENTICAL to finding large 3-free sets, that is, large sets that do not have arithmetic progressions of length 3.)
JASON: That is the dumbest problem I ever heard!. First off, chess is played on an 8x8 board. Secondly, only once while playing chess did I ever get my pawn to the end of the board to get a second queen, never mind getting n-o(n) queens (he didn't put it that way). And thirdly... who cares?
BILL: I don't suppose it would help to tell you that this relates to some deep mathematics of interest.
JASON: Of interest to who?
BILL: Uh... Never mind. (I thought about telling him that better bounds on the VDW numbers could get you $3000 dollars but decided not to.)I usually do better than this with very concrete easily grasped statements. But he doesn't care about this problem. And actually, why should he? (I am sure some of my readers don't care about this problem either.)