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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 …

Yes. Just take $\mathcal{A}$ to be $[\omega]^2$ together with the powerset of the even integers.

Here is a proof of Conjecture 1. Proof. We prove the contrapositive. Suppose that $G$ is not a cograph. Then $G$ has an induced subgraph $H$ such that $H \simeq P_4$. Let $V(H)=\{1,2,3,4\}$ and …

Yes, we may take the function to be $2\Delta$. Lemma. Every interval graph $G$ has an interval representation where all intervals have length between $1$ and $2\Delta$, where $\Delta$ is the maximum d …

Here are some upper and lower bounds. The paper On the chromatic number of some flip graphs proves that the chromatic number of $G_k$ is at most $4k-4$. Therefore, in every proper colouring of $G_k$ t …

This is a design theory question. You are asking about the existence of a Balanced Incomplete Block Design (BIBD). A $(v,k,t,\lambda)$-design is a collection of $k$-subsets (called blocks) of a $v$- …

No, this is false even in the planar case. Let $G=W_n$ be a wheel graph with $n \geq 6$. Deleting any edge of the outercycle yields a fan graph, which is not $3$-connected. On the other hand, contr …

Here is a short proof that at most $2n-2$ swaps are necessary. We proceed by induction on $n$. For the base case $n=1$, it is clear that no swaps are necessary. For the inductive step, let $X_1,X_2 …

This is false in general. Let $X=\{1, \dots, n+1\}$, $Y=\{n+1, \dots, 2n+1\}$, and $Z$ be any $(n+1)$-subset of $[2n+1]$ not containing $n+1$. Let $\mathcal{F}=\mathcal{A} \setminus \{X,Y,Z\}$. The …

According to Gallai’s question and constructions of almost hypotraceable graphs by Wiener and Zamfirescu, this is an open problem (see the beginning of Section 4). Note that a graph is $G$ hypotrace …

There is no such graph on $11$ vertices, but for all $n \geq 12$, there exists a thickness-$2$ graph with $6n-12$ edges. Both these results were proved by Boswell and Simpson in Edge-disjoint maximal …

Let $x$ be a vertex of $\mathcal{P}$. By Cramer's rule, there is an $n \times n$ matrix $C$ such that each coordinate of $x$ is an integer multiple of $\frac{1}{|\det(C)|}$, and the absolute value of …

Here is a counterexample for $n=5$. Partition the non-empty subsets of $\{1, \dots, 5\}$ into the singleton subsets and a sixth family containing all the other non-empty subsets. So, $m=6$ and $|\ma …

For an upperbound, the clique number of a $k$-critical graph is obviously at most $k$, and this is achieved by the complete graph $K_k$. There is no non-trivial lowerbound for the clique number, and …

This follows from Proposition 2.1 of the paper Many $T$ copies in $H$-free graphs by Alon and Shikhelman. Theorem (Alon and Shikelman) Let $T$ be a fixed graph with $t$ vertices. Then $ex(n,T,H)=\Ome …