Three steps towards Gyárfás’ conjecture

Gyárfás conjectured in 1985 that for all k, l, every graph with no clique of size more than k and no odd hole of length more than l has chromatic number bounded by a function of k, l. We prove three weaker statements: • Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; • For all l, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than l; • For all k, l, every graph with no clique of size more than k and sufficiently large chromatic number contains either a 5-hole or a hole of length more than l.