The Link Between Price Inflation & Unemployment
Yes there is. Obviously, the correlation is not perfect, or even close to being perfect, but in most cases there is.
Exceptions from the rule is when there is some form of negative supply shock, for example when certain Arab countries initiate an oil blockade, or when there is a dramatic drop in the exchange rate associated with a weaker economy.
But setting aside such examples of negative supply shocks, it generally holds true that higher inflation is associated with lower unemployment and vice versa.
But why is that? There are two possible explanations: either that higher inflation causes unemployment to drop, or that lower unemployment causes inflation to rise.
The first explanation is actually relatively uncontroversial among Austrian/ free market economists: when there for whatever reasons exist nominal wage rigidity and wages in some sectors are too high, causing unemployment, then higher inflation by reducing these excessive real wages can reduce unemployment. As the inflation solution has negative side effects, this doesn't necessarily mean that inflation is desirable, but it does mean that it has one positive effect.
The second explanation is more controversial, as that explanation is perceived as Keynesian and therefore rejected. Until recently I too rejected it, but having thought it through I realize that it could be partially true.
That is for two reasons. First of all, the Penn effect, which says that there is a positive correlation between income levels and price levels, something which in turn implies that higher growth will raise inflation. And as higher growth is associated with lower unemployment, it follows that lower unemployment will raise inflation. However, if there is a floating exchange rate (including managed float) it is possible that nominal exchange rate movements instead of relative inflation can accommodate the Penn effect.
And secondly, price inflation is not entirely determined by monetary inflation (money supply growth). It is also determined by money demand and the supply of goods and services. Regarding that last point, it should be noted that assuming at least some wage flexibility (and that is in fact a reasonable assumption, even though the assumption of full wage flexibility is not reasonable) then a higher rate of unemployment will mean that the price of labor will go down. Though that will mostly manifest itself in terms of a lower real wage, a cheaper price of labor will also induce capitalists to use this to lower prices with maintained margins., something which will mean lower price inflation.
It should be noted that while the above reasoning does provide theoretical support for the Philips Curve, it doesn't provide support for Keynesianism as they both involve structural supply issues.