In the company, employees are allowed to work more hours (unpaid), accumulate them, and take days off (paid as if they worked) when they have enough accumulated hours to make up for the day off. This is entirely voluntary and pretty much every employee is allowed to use this. The only condition is that your immediate superior, which is usually your team captain, is informed of it at least 3 work days in advance.
That specific person you are reporting to is also allowed to say no to a particular date, if for example there is an important meeting scheduled for that day, or some particular work-related event (a scheduled team-wide push-to-prod that might require people to be available if something goes wrong, for example). I'd say about 95% of such days off are approved instantly without any arguing whatsoever.
Company policy is extremely flexible about that point. This means that the "warn in advance and get approval" requirement is usually done verbally when you encounter the person, sometimes at lunch, or walking past their office or something like that. There is no paper or electronic trail, since this is mostly a "for everyone's convenience" policy, and it's not the end of the world if someone doesn't ask.
I have a few people reporting to me, so I get those demands pretty regularly. As you would expect from this kind of policy, most days off are taken on Fridays and Mondays, so even when not that many are used, they often fall on the same days as others. I have been the kind of guy who forgets those demands pretty fast, particularly if they were asked while I was doing something else. It happened quite a few times that, when the co-worker came back after their day off, I'd ask them "Hey, did you ask for permission for yesterday's day off?". I am aware that this was a problem on my end, and have fixed it.
I am now taking written notes of those demands every single time. If I can't send myself an email reminder right away, I use a piece of paper or my cellphone, and I am quite confident that I am not missing any anymore.
But there is this one guy who seems to have caught on early, and who often answered with "Yes, I asked you on day x, don't you remember?". The problem is that he seems to keep doing so, even now, when I am 99.9% certain that he, in fact, did not ask.
I have to mention that sometimes he does ask and I take notes and remember it. It just feels like sometimes he either forgets to ask, or decides it at the last minute, and chooses to take advantage of a weak point that isn't there anymore.
It is very hard to keep a paper trail of things that did not happen, for obvious reasons. It would be unreasonable to be the only guy in the building who only accepts such demands if they are asked in writing, and probably against our flexible rules. I have also truly forgotten such demands many many times in the past, so no matter how reliable my new technique is, I bet I don't have that much credibility if I try to argue that he didn't really ask.
I also want to specify that not a single time in the past has he taken such a day off without asking, that I would not have accepted if he has asked. So no problems at all have arisen from this behavior. I just fear the day it actually becomes a problem, which seems to only be a matter of time until it happens.
So my question is: How can I make this risky situation stop, and in the event that I can't, how can I make sure that if any real problem ever arises from it, I have not held responsible because of my past behavior?