It seems pointless to argue over what might exist if we do not have a clear understanding of what it really means to exist;
Existence == A set of entities which share the property of mutual affect or potential to cause change; a realm of mutual affectance.
1) Detectable Empiricism - We decide that something exists only when we detect that something is having affect. All of our senses function based on the affect that something else has upon them. We use equipment to increase our sensory ability, but still if nothing affects the equipment in any way, we declare that nothing was there.
2) Common Usage - In reality, people are already using the word "exist" to mean this definition. They often never think about it, but in every case, the person really means that something existing means that it has the potential to affect something; being seen, touched, smelled, or detected in some way even if not already doing so.
3) Support from Science - Science concluded long ago that in reality all things have at least some minuscule affect on all other things. Often this is more of a chain effect than a direct effect.
4) Rational relevance - If something has truly no affect on anything whatsoever, we really don't care if it exists in any other sense anyway. We can propose trillions of things that might exist but don't have affect. What would be the point? It would be a waste of mind time.
Existence comes in realms or sets. A realm of existence includes all mutually affecting entities such as the realm of physical existence. A different realm of existence can be conceived wherein all items have mutual affect but do not affect those of the physical realm. In such a case, the items would not physically exist.
The realm of concepts is the set of ideas wherein each idea has affect upon the others yet no perfect concept can have physical existence. A perfect square cannot physically exist but the concept of a perfect corner and of a perfect line affect what a perfect square is. Each non-physical perfect concept affects the other perfect concepts.
Concepts involved in the definition;
Realms and sets