\( \newcommand{\fnorms}[1]{{\lVert#1\rVert}^2_F} \newcommand{\pr}{\mathrm{P}} \newcommand{\card}[1]{\lvert#1\rvert} \newcommand{\RR}{\mathcal{R}} \)
[RS] Let \(A\) be a \(d\)-by-\(2d\) random matrix with iid standard Gaussian entries. Then \[ \pr\Bigl(\min_{S\subseteq [2d], \card{S} = d} \sigma_d(A_S) \geq e^{-\Omega(d)}\Bigr) \leq e^{-\Omega(d)} \] (where \( A_S\) is the subset of columns of \(A\) indexed by \(S\) and \( \sigma_d(A_S) \) is its \(d\)th smallest singular value).
[LR] Consider a set of \(2n+5\) points on the plane in general position. Then the number of conic sections going through 5 points and splitting the rest of the points in half is \( \binom{n+2}{2}^2 \).
[DR] [DRVW] Let \(A\) be an \(m\)-by-\(n\) real matrix. Let \(1 \leq k \leq \min\{m, n\}\). Let \(A_k\) be the best rank-\(k\) approximation to \(A\). Then there exists a polynomial time computable matrix \(C\) formed by \(k\) columns of \(A\) such that \[ \fnorms{A - C C^+ A} \leq (k+1) \fnorms{A-A_k} \] (where \( C C^+\) is the projection matrix onto the columnspace of \(C\)).
[FGRV] The union of two random spanning trees of the complete graph on \(n\) vertices has constant vertex expansion with probability \(1-o(1)\).
[R] For \(K \subseteq \RR^d \), let \(V_K\) denote the expected volume of the convex hull of \(d+1\) iid random points from \(K\). The statement: "for all pairs of compact convex sets with non-empty interior \(K \subseteq L \subseteq \RR^d\) one has \(V_K \leq V_L\)" is true for every \(d \in \{1,2\}\) and false for every \(d \geq 4\).
(The case \(d=3\) was resolved in [RR], negatively.)