$$\newcommand{\fnorms}{{\lVert#1\rVert}^2_F} \newcommand{\pr}{\mathrm{P}} \newcommand{\card}{\lvert#1\rvert} \newcommand{\RR}{\mathcal{R}}$$
[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.)