If you want to get a job in this economy, it's becoming increasingly clear that you better also be thinking about getting an education beyond just a high school degree.

The government reported Friday that the unemployment rate for college graduates fell slightly in May, to 3.9 percent from 4 percent a month earlier. For people with just a high school degree, the unemployment rate increased to 8.1 percent, from 7.9 percent a month earlier. The overall unemployment rate was 8.2 percent in May, the Bureau of Labor Statistics said. That reflects even higher rates of unemployment among those who don’t even have a high school degree.

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In the past six months or so, Downing said he’s started to see such high demand for college graduates, particularly in fields such as engineering and information technology, that some candidates are fielding multiple job offers.

That’s a situation he said he and his partners hadn’t encountered in three years, and that is catching picky employers off guard.