Defect and transference versions of the Alon–Frankl–Lovász theorem

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.