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The number of $m$-subsets of {$1,2,\ldots,n$} with distance at least $k$ between any pair is $n - (k-1)(m-1) \choose m$. Proof: for any subset of size $m$ of the first $n-(k-1)(m-1)$ integers, you ca …

We will let $k_1=k$ and $k_2=n-k$, and $\tilde{c}=c/2$. I believe the answer is then $\frac{k_1 + k_2}{\tilde{c}} {k_1-1 \choose \tilde{c}-1} {k_2-1 \choose \tilde{c}-1}$. I have no idea whether th …

Having screwed up the answer by getting the wrong answer for a really simple calculation the first time, I'm now going to try to redeem myself. First, to make things easier, let each point be present …

Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan) 1 1 3 2 10 15 6 40 105 105 24 196 700 1260 945 which is defined by $S(i,j) = i\ S(i …

Here's the start of a proof. We will say that two arcs overlap twice if their union covers the entire circle. First, note that you can assume that there are no two red arcs that overlap twice. Proo …

EDIT: I see that Steve has more or less the same construction above. I should have read his answer more carefully before I posted. I don't believe it's true. Let's say you have a square polyomino wit …

The quantum PCP conjecture (nobody has proved it, so you can't call it a theorem) is possibly (there are a few different ways of generalizing the classical PCP theorem to the quantum regime, and I don …

This is the simple stochastic games problem, but for only one player, and there is a polynomial-time algorithm for it based on linear programming, which is described in Anne Condon's paper "On Algorit …

You can get half of the elements small. Let $e_k$ be the element $(0,0,\ldots,0,1,0, \ldots 0)$ with a single $1$ in the $k$th position. Let $v$ be the element $(1,1,1,1,1,\ldots,1)$. Now, consider th …