Expander graphs and sieving in combinatorial structures

Graphes expanseurs et crible dans les structures combinatoires

Florent Jouve (1) , Jean-Sébastien Sereni (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 ORPAILLEUR - Knowledge representation, reasonning

We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (non-empty) subsets summing to zero, or that a random coloring of the edges of a complete graph contains a monochromatic triangle.

Mots clés: Groups; Random Walks; Graphs; Sieve

Domaines: Théorie des groupes [math.GR]; Combinatoire [math.CO]

Florent Jouve, Jean-Sébastien Sereni. Expander graphs and sieving in combinatorial structures. Journal of the Australian Mathematical Society, 2018, 105 (1), pp.79--102. ⟨10.1017/S1446788717000234⟩. ⟨hal-00693334v3⟩