Defect and transference versions of the Alon-Frankl-Lovasz theorem
Lior Gishboliner, Stefan Glock, Peleg Michaeli, Amedeo Sgueglia
TL;DR
This work extends the Alon–Frankl–Lovász theorem from dense complete hypergraphs to sparse random hypergraphs by proving a defect version for almost complete hypergraphs and a transference theorem for $\mathbb{G}^{(r)}(n,p)$ with $p \gg n^{-r+1}$. The defect proof leverages extremal set theory, notably the Erdős matching conjecture, together with a novel almost-cover (shadow) argument to extract a large monochromatic matching. The transference result then follows via a multicolor sparse hypergraph regularity lemma, transferring the cluster-hypergraph conclusion back to the original graph and yielding a matching of asymptotically the same size as in the dense AFL setting. A discrepancy version for perfect matchings at higher density thresholds is also obtained, and the paper closes with discussion of extremal colourings, stability questions, and open directions.
Abstract
Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac{n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon--Frankl--Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.