Is there any connection between stagflation and Hayek's philosophies?
Stagflation is the situation in which an economy experiences both high unemployment (stagnation) and inflation.
There are people who would argue that the existence of stagflation in the 1970s in the United States proved that Hayek was right and that Keynes was wrong. Keynes argued that the government could keep economic growth going (and, with it, low unemployment) so long as it was willing to accept some inflation. The stagflation of the 1970s, it is said, disproved this idea. It showed that inflation could happen even as unemployment went up.
You can argue that this proves that Hayek was right. It proves that the economy is better off without government intervention because a free market economy would correct itself and stagflation would not be able to persist.