Short proofs in combinatorics, probability and number theory II

We give a quintet of proofs resulting from questions posed by Erdős. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a “fewnomial” version of the Erdős–Turán discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a{k}^2$ is prime for all $k\le \sqrt{n\mathrm{/}a}$ coprime to $n$ (for fixed $a\in {\mathbb{Z}}_{+}$). Each proof is due to an internal model at OpenAI.