Reflections on a Problem of Erdős and Hajnal

The problem I am going to comment reached me in 1987 at Memphis in a letter of Uncle Paul. He wrote: 'We have the following problem with Hajnal. If G(n) has n points and does not contain induced C4, is it true that it has either a clique or an independent set with n 10 points? Kind regards to your boss+ colleagues, kisses to the E -s.' E.P.' After noting that E have been used in different contexts I realized soon that ~ is a good E (in both senses since I have three daughters). About a month later Paul arrived and said he meant Cs for C4. And this minor change of subscript gave a problem still unsolved. And this is just a special case of the general problem formulated in the next paragraph.