MathOverflow

Suppose we measure the length of a proof by the number of logical steps, symbols, lines, or pages used.
If a theorem has at least one valid proof, does there necessarily exist an absolutely minimal proof within a fixed logical system?
Or is it possible that every proof could always be shortened further through a completely different method or perspective, while still remaining fully rigorous and logically complete?...

I am exploring the hypothesis that the difficulty of the P vs NP problem might stem from a structural analogy to the Liar Paradox and Gödel’s Incompleteness Theorems, rather than solely from a lack of efficient algorithmic ingenuity.​If a decision problem...

Given a set $A$ and a variety (in the sense of universal algebra) $\mathbb{V}$, let $\mathbb{V}(A)$ be the set of all $\mathbb{V}$-algebras with underlying set $A$. Let $\mathbb{G}$ be the variety of groups. Given a group $G$, I'll write "$\underline...

I would like to compile a list of questions about graphs that have a non-trivial analogs for bipartite graphs.
Let me give the following examples:
Cycle vs Even cycle. Most questions about cycles in graphs have analogs in even cycles for bipartite graphs. For instance, it is trivial to show that a bipartite graph on an odd number of vertices cannot have a Hamilton cycle. In such a case the bipartite...