We prove that $N$ points in $\mathbb{R}^3$ determine at least $N^{2/3-o(1)}$ distinct distances.
Let $h\geq 3$ be an integer and $0<\alpha<1/h$. In this paper, we prove that there exists a minimal asymptotic basis of order $h$ with asymptotic density $\alpha$. This solves an open problem posed by Erd\H{o}s and Nathanson in 1988.
We present an exposition of a proof, discovered by GPT-6 Astra, of the Erd\H{o}s--S\'os Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.
We give a Green-function proof of the stable Bernstein theorem in $\mathbb R^7$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.
Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{89}{90} + o(1)}$.
We prove the Erd\H{o}s--Chv\'atal simplex conjecture across its entire parameter range. If $2\le d<k$ and $(d+1)k\le dv$, then every $d$-simplex-free family of $k$-subsets of $[v]$ has at most $\binom{v-1}{k-1}$ members, with equality only for a full star.
In 1985, Erd{\H{o}}s, Frankl and F{\"u}redi conjectured that the threshold $T(r)$ for the existence of an $r$-cover-free family with more members than points satisfies $T(r)=(1+o(1))r^2$. We confirm this conjecture by proving $T(r)\ge r^2+r+2$ for every $r\ge1$.
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