# Relation between complements and substitutes (for multiple goods)

I am a little bit curious about the following problem:

If we have multiple goods (at least 3 or more)... And we know that $$x_1$$ and $$x_2$$ are substitutes and $$x_1$$ and $$x_3$$ are also substitutes, does it imply that $$x_2$$ and $$x_3$$ are substitutes as well? I would say so, just intuitively: If I do not mind consuming $$x_1$$ instead of $$x_2$$ and I do not mind consuming $$x_3$$ instead of $$x_1$$, I would also have to be indifferent (utility-wise) between $$x_2$$ and $$x_3$$ through the mentioned path...

BUT are there any robust conditions, for example in case of 4 goods and more?

• $$x_1$$ and $$x_2$$ substitutes
• $$x_1$$ and $$x_3$$ complements
• $$x_3$$ and $$x_4$$ substitutes

Can I say in such a case that $$x_2$$ is also a complement with $$x_3$$ or $$x_1$$ is a complement to $$x_4$$ etc...?

What would it be in case of imperfect substitutability? Let's say $$x_1$$ and $$x_2$$ have elasticity of substitution equal to 3, what would it imply about elasticity of substitution for $$x_3$$ and $$x_2$$ etc...? Is there some math behind it what conditions must hold in these cases?

Thank you very much!