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The answer to both of your questions is yes. As suggested by the edit, consider the graph $G(A)$ whose vertex set is the set of non-zero entries of $A$, and where two entries are adjacent if they …

One possible generalization comes from the graph minors project of Robertson and Seymour. In particular the notion of tree-width is in some sense dual to the notion of a bramble (I will define this i …

Here is a proof of a very weak upper bound for $l(k)$. Consider the colouring of $2^S$ where each set is coloured by its size (mod 1000). A good monochomatic sequence must consist of sets of the sam …

Edit. The answer below is incorrect, but I'll leave it here for others to avoid the same pitfall. Here is a formula that I would rate $\epsilon$ less than horrible. Let $M=\{1^{a_1},\dots,m^{a_m}\ …

I assume what is meant is whether it is always possible to choose a size $|\mathcal R|$ subcollection $\mathcal U$ of $\mathcal T$ and elements $e_U \notin U$ for each $U \in \mathcal U$ such that $\{ …

To address the updated question, a family of subsets of $[n]$ is called $k$-Sperner if it does not contain a chain of length $k+1$. By taking all sets whose size lies in the middle $k$ values of $[n] …

The answer to Question 1 is negative. Let $G=\{\{1, \dots, N+1\}\}$ and $H$ consist of all subsets of $\{1, \dots, N+1\}$ of size $N$. If $K$ is a distinguisher for $G$ and $H$, then for each $i \in …

No, this is not possible. Towards a contradiction, suppose such a partition $S_1, \dots, S_{2n-1}$ of $E(K_{2n})$ exists. I first claim that each $S_i$ must be a matching. If not, then some vertex $ …

The characterization given by Noam Elkies is correct. Indeed, in this paper Bryant, Horsley and Pettersson prove the stronger result that if $n$ is odd, and $m_1, \dots, m_t$ are such that $3 \leq m_ …

Note that what you defined is simply the dimension of the nullspace of the adjacency matrix of $G$. This is usually called the nullity $\eta(G)$ of $G$. Apparently, this parameter is significant in …

Yes. Let $C_n$ be the cycle on $n$ vertices. The size of a largest induced matching in $C_n$ is exactly $\lfloor \frac{n}{3} \rfloor$, since we can take at most every third edge of $C_n$. On the oth …

Matrices that contain at most two non-zero entries per column are called frame matrices. The matroids representable by frame matrices (over a finite field $\mathbb{F}$) are in fact a fundamental clas …

I am aware of the paper, but I am not sure that MO is the right forum for this sort of question. Nonetheless, let me try to provide some information in as neutral a manner as possible. Note that t …

The answer is no. A counterexample is the triangular prism graph. Up to symmetry, there is a unique 5-cycle and a unique 6-cycle in the triangular prism and both these cycles have chords. Hence the …

The number $m_n$ of matroids on $n$ elements is an interesting example, since a priori, it is unclear how fast $m_n$ grows. Knuth (1974) showed that $\log \log m_n$ is at least $n- \frac{3}{2} \log n …