How BIG must a ASTEROID be to DESTROY MANKIND?

More than 10 miles in diameter. Maybe more than 20. We actually don't know for sure. There are asteroids much bigger than that, but they are all safely in stable orbits between Mars and Jupiter. A few asteroids are in Earth crossing orbits, But almost all of those is smaller than 10 miles in diameter. A 1 mile asteroid could cause a lot of damage, like take out a big area, but not cause an extinction event. We are pretty sure we have discovered all the asteroids big enough to cause extinction. The search goes on for the smaller ones. All the big ones seem it be safely missing us and knowing when and where a smaller one would hit could save millions of lives by evaculating the area. But evacuating all of, say Europe, would be WAY more expensive than mounting a mive, emergency, cost is no object, bigger than the Apollo program effort to deflect it.