What are some red flags that even the professors cannot help in the admission?

According to Wolfgang Bangerth's answer to Can a professor influence the graduate admissions committee if he or she is interested in a particular student?:

If a professor sends us a note that they'd like to have a student admitted and state that they will take on their adviser role, we [the adcom] will only not admit that student if there are any red flags in the file.

I want to elaborate this more. What are the red flags that he mentions?

I know that they will look at these things: the transcript, the LORs, the SOP, the GRE and TOEFL scores. What is the threshold of them? At the 50% bottom of the applications or what? Is the threshold different from top schools and low tier ones?

I have applied to a top school, to a leading professor, who says that I "would be a terrific addition" to the group. If I can avoid the red flags, how much chance that I will get an admission?

Answer

What are the red flags that he mentions?

In general, a red flag is anything that makes the application look inappropriate or weak in comparison with other applications. This is a vague term, as you've recognized. It can mean something a little different to each committee member. (I've been in discussions in which one committee member thought something was clearly a worrisome sign while another was completely unbothered by it.)

In practice, when you first look at a group of applications they come across like this. There's a small group that should obviously be admitted, and a large group that don't look like good choices. The latter are the ones with red flags. Then there's a group that look like they could plausibly be admitted, but there are too many of them so further distinctions have to be made. That's the group where support from a potential advisor would be most helpful.

I have applied to a top school, to a leading professor, and they says that I "would be a terrific addition" to the group. Would this mean that if I'm trying to avoid the red flags, 100% that I will get an admission?

It's completely unclear. First of all, it depends on how the admissions committee works. For example, it sounds like support from an advisor helps much more in Wolfgang Bangerth's department than in mine. Without knowing where you are applying, it's impossible to say what will happen in your case.

Furthermore, who knows what this professor's statement means? Maybe they tell lots of people that they would be terrific additions. Maybe you're special but someone even more impressive has also applied. It's not worth spending a lot of time trying to interpret statements like this, since you just don't have enough information to say with any confidence what they mean.