To the comments from a few months ago,
1. Bakaprince didnt even notice Sei. He only saw one person. Convenient but idiocy is his characteristic so lets just leave this be.
2. Sei wanted to 'meet' with the saint but she doesn't want to get involved with the prince. Also, she heard that the saint was being taken care of so there is no problems there.
3. Sei's food is not widely spread in the castle. The research area is quite some distance away. Saint has small appetite because the food doesn't taste good. This gets better later.
4. There is only one saint (at least in this summoning). The saint is Sei. The other girl, Aria i think she was called, is a mage.
5. Bakaprince thought Aria was the Saint because she was dressed well and was pretty(ier). At the moment of summoning, Sei was overworked, sleep deprived with terrible complexion and black rings under her eyes. Everyone else with common sense had split opinions.. BUT...
6. They ignored Sei because she 'collapsed' (read : falls asleep) during the day, soon after she was summoned. Her anger and frailness made them afraid to approach her (lest she leaves and they become responsible). It was 'daytime' when they were summoned but Sei was going home from overtime work.
All this information is from the WN/LN (+ some common sense)