This page collects my preprints, publications, notes, and slides.
Preprints
with Zixiang Xu and Yang Zhou.
In 1983, Kusner conjectured that the largest equilateral set in
\(\mathbb{R}^{n}\) with metric \(\ell_{p}\) has cardinality \(n+1\)
when \(1\lt p\lt\infty\) and \(2n\) when \(p=1\).
This conjecture was proved only in the isolated cases \(p=2\) and \(p=4\),
and was disproved when \(1\lt p\lt 2\). The best general upper bound
\(O_p(n^{\frac{2p+2}{2p-1}})\) is due to the celebrated work of
Alon and Pudlák, GAFA, 2003. Our main contributions include:
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We prove Kusner's conjecture for every dimension \(n\ge 1\) when
\(2\le p\le 4\). More generally, for every integer \(k\ge 0\)
and every \(p\in[4k+2,4k+4]\), every equilateral set in
\(\mathbb{R}^{n}\) with metric \(\ell_p\) has cardinality at most
\((2k+1)n+1\). On the complementary intervals
\(p\in(4k,4k+2)\) with \(p\ge 1\), we obtain the almost linear bound
\(O_p(n\log n)\).
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We also consider the analogous problem on the torus \(\mathbb{T}^n\),
recently initiated by Alon, where the cyclic distance makes the problem
substantially more delicate than in \(\mathbb{R}^n\). We prove the almost
linear bound \(O_p(n\log n)\) for \(1\le p\le 2\) and
\(O_p(n^{\frac{3}{2}-\frac{1}{p}})\) for every fixed real \(p\gt 2\),
improving Alon's bounds \(O_p(n^{2+\frac{2}{\lfloor p\rfloor}})\)
for all finite \(p\ge 1\).
Preprint, 2026.
Publications
with Jack H. Koolen and Akihiro Munemasa.
We construct a \(2\)-distance set with \(277\) points in the
\(23\)-dimensional Euclidean space \(\mathbb{R}^{23}\), whose pairwise
distances between distinct points are \(2\) and \(\sqrt{6}\).
Appeared in Discrete & Computational Geometry, 2026.
with Jack H. Koolen.
Tan, Koolen, and Xia conjectured that connected co-edge-regular graphs
with four distinct eigenvalues and fixed smallest eigenvalue, when having
sufficiently large valency, belong to two different families of graphs.
In this paper we construct two new infinite families of connected
co-edge-regular graphs with four distinct eigenvalues and fixed smallest
eigenvalue, thereby disproving their conjecture. Moreover, one of these
constructions demonstrates that clique-extensions of Latin square graphs
are not determined by their spectrum.
Appeared in Electronic Journal of Combinatorics, Volume 32,
Issue 3, 2025.
with Jack H. Koolen, Chenhui Lv, and Qianqian Yang.
We combine a Bose–Laskar type argument with Hoffman theory to obtain
structural results for \(\mu\)-bounded graphs with fixed smallest
eigenvalue, avoiding the Ramsey-theoretic approach used in earlier work of
Koolen, Yang, and Yang. Since local graphs of distance-regular graphs are
\(\mu\)-bounded, we apply our results to distance-regular graphs with
classical parameters \((D,b,\alpha,\beta)\). In particular, we show that
\(\alpha\) is bounded above by a cubic polynomial in \(b\) whenever
\(D\ge 9\) and \(b\ge 2\), and that \(\alpha\le 2\) when \(b=2\) and
\(D\ge 12\).
Appeared in European Journal of Combinatorics, Volume 136, 2026.