Tianhong Kan | Searcharxiv

arXiv2026

Stability and exact extremal colourings for the Alon--Frankl--Lovász bound

The Alon--Frankl--Lovász theorem states that every $q$-colouring of the edges of the complete $r$-uniform hypergraph on $n$ vertices contains a monochromatic matching of size $\lfloor(n+q-1)/(r+q-1)\rfloor$. For fixed uniformity $r$ and number of colours $q$, we prove a stability theorem: every colouring whose largest monochromatic matching has size at most $n/(r+q-1)+o(n)$ differs from a threshold extremal colouring on only $o(n^r)$ edges. We also give a necessary and sufficient structural characterization of all extremal colourings for sufficiently large $n$. The proofs combine rigidity of fractional covering weights with an extension argument for prescribed matchings. These results answer, for sufficiently large orders, the stability and extremal-colouring questions posed by Gishboliner, Glock, Michaeli and Sgueglia.

math.CO↗

arXiv2026

Ore-degree conditions for hypergraph matchings

Let $σ_k(H)$ be the minimum sum of the vertex degrees over all non-edges of an $n$-vertex $k$-uniform hypergraph $H$. For every $k\ge3$, $n\geq ks$ and sufficiently large $n$, we establish a sharp Ore-type degree condition that forces a matching of size $s$ in $H$. Specifically, we prove that if \[ σ_k(H)>\max\left\{ k\left(\binom{n-1}{k-1}-\binom{n-s}{k-1}\right), (k-c)\binom{ks-2}{k-1}\right\}, \] then $H$ has a matching of size $s$, where $c=n-ks+1$. In particular, this proves a conjecture of Balogh, Palmer and Raeisi for $k\geq 3$ and sufficiently large $n$.

math.CO↗