Some problems on induced subgraphs.

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. [ABSTRACT FROM AUTHOR]