Year: 2026
Type: article
Abstract: Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any q q $q$ -colouring of the edges of the complete r r $r$ -uniform hypergraph, there exists a monochromatic matching of size left floor StartFraction n plus q minus 1 Over r plus q minus 1 EndFraction right floor ⌊ n + q − 1 r + q − 1 ⌋ $\lfloor \frac {n+q-1}{r+q-1}\rfloor$ . In this paper, we prove a transference version of this theorem. More precisely, for fixed q q $q$ and r r $r$ , we show that with high probability, a monochromatic matching of approximately the same size exists in any q q $q$ -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.
Institutions: University of Toronto, University of Passau, University of Oxford
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7141350622