The straight line

My backpack got stolen. It had just been urinated on by a cat, and was evidently beyond salvation (it was pretty beat up anyway). I emptied it and left it to stink outside of the pizzeria we’d decided to eat at, planning to take it to the next garbage bin I saw. But somebody nabbed it! I hope they don’t make the mistake I did of wearing the backpack – my nice winter coat now has a faint whiff of un-neutered male cat piss.

We were discussing Ramsey’s function (R(k) = the smallest number of people for which you can guarantee that either k people among them all know each other or k people all are strangers to each other) with Vadim. Its values are known to lie between and , which is obviously quite a large gap. But how large? Vadim immediately said the difference is exponential, but it wasn’t so obvious to me. Eventually he convinced me. It then occurred to us that it’s in fact essentially ; given , the difference between the gap and the larger exponential function relative to the larger function goes to zero, . So when you subtract a smaller exponential function from a larger exponential function, you’ve basically subtracted nothing. Which is really a stupid thing to notice because it’s true even of polynomial functions (but not linear functions).