Why there is no open source anti-virus? And what problems that will associate with open source?
Your premise is wrong, there is: ClamAV.
Edit: Assuming that there are indeed surprisingly few open source antivirus tools, then this could be due to the fact that people who are competent enough to build them often don't believe in antivirus. This is just speculation on my part but the arguments against antivirus are solid and you can find many articles on it.
Why there is no open source anti-virus ?
There were a few open source anti virus programs back then. I believe all of them have gone extinct except ClamAV (very old though)
Make no mistake, there are still a lot of free anti virus programs out there. However, they aren't open source software.
Anti virus programs need to be constantly updated for virus definitions. Having an outdated anti virus software is as good as not having any. Creating and providing constant support to an updated open source software is a lot of work. There may be a lot of perspectives why many of them aren't around. May be they haven't found a sponsor or may be the people who sell paid ones don't want open sources to bite their tails(just saying..)
ClamAV has already been mentioned. As to why there aren't more open source offerings:
It takes a lot of resources to collect samples, analyze them quickly, and push out updates. Beyond that, the algorithms and scanning methods in the code itself have to be continually updated as malware techniques evolve. Who's going to fund that and give it away?
There are a number of free offerings, but they tend to be ad supported, or try to upsell you to a paid product. The one exception is Windows Defender, which Microsoft seems to offer as a baseline protection to fend off the "Windows is insecure" reputation.