Defect and transference versions of the Alon-Frankl-Lovasz theorem

Abstract

Confirming a conjecture of Erd\H{o}s on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lov\'asz 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--Lov\'asz 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 Erd\H{o}s matching conjecture.