If while fasting, the gums bleed and the person swallows the blood, would it break the fast?
If a person does not know about this issue and swallows the blood unintentionally, then it would not affect the fast.
However, if a person knows that swallowing anything breaks the fast but still intentionally swallows the blood, then it would break his fast and he would need to offer qada (قضاء) for it.
Laqeet bin Sabarah reported that he asked:
O Prophet of Allah صَلَّى اللّٰهُ عَلَيْهِ وَسَلَّم ! Tell me about Wudu (وضو)
He said: Perform wudu thoroughly, run fingers between your fingers and snuff water into your nose thoroughly unless you are fasting.
Sunan Abi Dawud:142
This hadith shows that if a person is able to spit out while knowing that swallowing it would break his fast yet he swallows it intentionally, then it would break his fast.
Allah Has created two paths to reach stomach; through mouth and through nose. So, anything that reaches the stomach through nose or mouth intentionally would break the fast.
واللہ اعلم بالصواب