Abstract
We discuss some problems related to induced subgraphs. (1)A good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length 3 or 4.(2)The perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect if it induces a perfect graph.)(3)Graphs in which the difference between the chromatic number and the clique number is at most one for every induced subgraph of the graph.(4)A weakening of the notorious Erdős–Hajnal conjecture.(5)A conjecture of Gyárfás about the χ-boundedness of a particular class of graphs.
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