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Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, …

Suppose you are trying to prove result $X$ by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result $X'$ (that you don't really care about) by induction. This h …

4-colour Theorem. Every planar graph is 4-colourable. This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because …

Here is an example from polyhedral theory. A matrix $A \in \mathbb{R}^{n \times m}$ is totally unimodular, if every square submatrix of $A$ has determinant $1, -1,$ or $0$. Totally unimodular matric …

To answer the question it is important to disentangle the proof as follows. Theorem 1. Every minimum counterexample to the 4CT is an internally 6-connected triangulation. Theorem 2. If $T$ is a min …

Here is a fairly straightforward proof which also proves various generalizations of your problem. Choose $c,d$ such that $\phi_2(c,d) \neq 0$. If no such $c,d$ exist, then $f$ is identically $0$ and …

I'll just remark that the fact that every surface has a finite number of excluded minors (and also topological minors) does not require the full strength of the Graph Minors Theorem. Indeed, the proo …

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, …

As others have mentioned, the answer to your title question is strictly speaking no. With regards to your other questions, it has been proven that it is undecidable to compute the excluded minors for …

Your question is essentially about extension complexity. In general, the extension complexity of a polytope $P$ is the minimum number of facets over all polytopes $Q$ which project to $P$. You are i …

If the coefficients are non-negative then you can always do it with at most two integer evaluations. That is, $P$ and $Q$ are equal if and only if $P(1)=Q(1)$, and $P(P(1)+1)=Q(Q(1)+1)$. Update. …

Using the Combinatorica package in Mathematica, the command NumberOfGraphs$[p,q]$ returns the number of non-isomorphic graphs with $p$ vertices and $q$ edges. If you want to implement this yourself, …

Here is a nice example due to Bjorn Poonen, which I have taken from this survey paper of Bruhn and Schaudt. It is motivated by the following observations. Let $\mathcal{A}$ be a union-closed family. …

More generally, it is a classic result (independently due to Kostochka and Thomason) that minimum degree $(\alpha+o(1))n \sqrt{\log n}$ suffices to force a $K_n$ minor, where $\alpha$ is an explicit c …

There is indeed a Menger's theorem for matroids first proven by Tutte. The reference is Tutte, W. T., Menger’s theorem for matroids, Journal of Research of the National Bureau of Standards—B. M …