Abstract In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with q 3 q ≥ 3 spins. The model is defined on the complete graph K₍₌ K Nm, whose vertex set is equally partitioned into m components of size N. For a configuration: \1, , Nm\ \1, , q\, σ: 1, ⋯, N m → 1, ⋯, q, the Gibbs measure is defined by ₍, () = 1 Z₍, (N ₕ, ₖ =₁^Nm J (v, w) 1\ (v) = (w) \), μ N, β (σ) = 1 Z N, β exp β N ∑ v, w = 1 Nm J (v, w) 1 σ (v) = σ (w), where Z₍, Z N, β is the normalizing constant, and >0 β > 0 is the inverse-temperature parameter. The interaction coefficient is J (v, w) = \{ array{ll 11+ (m-1) J & if v, w are in the same component, \\ J1+ (m-1) J & if v, w are in different components, array. } J (v, w) = 1 1 +
Kyunghoo Mun (Fri,) studied this question.