A class of graphs is hereditary if it is closed under isomorphism and induced subgraphs. A class scriptG of graphs is χ‐bounded if there exists a function f:N→N such that for all graphs G∈G, and all induced subgraphs H of G, we have that χ(H)≤f(ω(H)). We prove that proper homogeneous sets, clique‐cutsets, and amalgams together preserve χ‐boundedness. More precisely, we show that if scriptG and G* are hereditary classes of graphs such that scriptG is χ‐bounded, and such that every graph in G* either belongs to scriptG or admits a proper homogeneous set, a clique‐cutset, or an amalgam, then the class G* is χ‐bounded. This generalizes a result of [J Combin Theory Ser B 103(5) (2013), 567–586], which states that proper homogeneous sets and clique‐cutsets together preserve χ‐boundedness, as well as a result of [European J Combin 33(4) (2012), 679–683], which states that 1‐joins preserve χ‐boundedness. The house is the complement of the four‐edge path. As an application of our result and of the decomposition theorem for “cap‐free” graphs from [J Graph Theory 30(4) (1999), 289–308], we obtain that if G is a graph that does not contain any subdivision of the house as an induced subgraph, then χ(G)≤3ω(G)−1.