1. ## V-sentence

can we define a V-sentence phi such that phi has arbitrarily large finite models and , for any finite model G , |G| is even . and then finding a finite graph G such that |G| is even and G doesnot model the sentence phi that I mentioned above.

2. Could you say what a V-sentence is?

3. Let V be a vocabulary. A V-formula is a formula in which every
function, relation, and constant is in V. A V-sentence is a V-formula that is a
sentence.
example about it the V-sentence ∀y∃x(1+x·x = y).

4. @Mike12
It appears to me, that you are using some sort of highly specialized text material.
It seems to involve both graph theory and formal logic.
I think that you need to be more detailed in writing up the questions.

5. the question is to define a a V-sentence phi such that phi has arbitrarily large finite models and , for any finite model G , |G| is even . and then find a finite graph G such that |G| is even and G doesnot model the sentence phi .

6. Let V be a vocabulary. A V-formula is a formula in which every
function, relation, and constant is in V.
And what is the vocabulary V in this particular problem? If V is completely arbitrary, then what is the goal of emphasizing that $\phi$ must only use V? Every formula is in some vocabulary. This is similar to saying, "Find a person from a country": every person is from some country. Also, later you are talking about a graph that is or is not a model of $\phi$. This means that V must have relations and/or functions to talk about graphs, so V is not completely arbitrary.

Edit: Maybe "a person from a country" is not a good example. I mean, "from a nation" rather than "not from a city".