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Not a good example, but it might hint for a better one. Take two integers $N=2k+1,M$. Let the basis sets be $\{x,x+1,...,x+k\},1\leq x\leq N$, taken modulo $N$ (identify $N\equiv 0$) and $\{1,2,...,N, …

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist …

It is well known that in a perfect graph, there exists a clique which intersects all maximum independent sets (stable set), as in the proof of weak perfect graph theorem. So I want to know is the str …

We have $A\subset A^+\cap D$ and $(A^+\cap D)^+=A^+$, so we can assume $A=A^+\cap D$, let $V=A^+$ the problem turn into $\frac{|V|-|V\cap U|}{|P|-|U|}=\frac{|V\cap(P-U)|}{|P|-|U|}\leq\frac{|V\cap U|}{ …

Let $V(G),E(G)$ be the vertex set and edge set of $G$. Take $n>|V(G)|$, and let $K_n$ be the complete graph on $n$ vertices. We take $H_1=K_n$ and $H_2=K_n\times G$ (the Cartesian product of graph). L …

The forbidden minors $\mathcal{F}$ are empty graph on $V(G)+1$ vertices and all graphs have at most $V(G)$ vertices and are not minor graphs of $G$. We easily have $\mathcal{F}$ is finite, the empty g …

Yes, let $M$ be the adjacency matrix of $Q_k$ over $F_2$, $V$ be the $F_2$-vector space with basis $(v_u)$ indexed by the vertex set of $Q_k$, denote $v_A=\sum_{u\in A}v_u$. We see that $Mv_A=v_{F(A)} …

Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$) I conjecture that there are only $4$ t …

Let $F_n$ be a set of all irreducible fraction $\frac{p}{q}$ such that $0<\frac{p}{q}\leq 1,1\leq p,q\leq n$ and for $i\in \{1,...,n\}$, $D_i$ be subset of $F_n$ which contain all irreducible fractio …

Let $G$ be a group. Consider the set $P(G)$ of all pairs $(H,N)$ of subgroups of $G$ such that $N$ is a normal subgroup of $H$. Consider the relation $\leq_G$ on $P(G)$ defined as follows: $(H,N)\leq_ …

Cross-posted from MSE: https://math.stackexchange.com/questions/4775193/for-which-set-a-alice-has-a-winning-strategy Alice and Bob are playing a game. They take an integer $n>1$, and partition the set …

Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which bel …

Let $G$ be a finite group, $G',H$ be its subgroups and $H'=G'\cap H$. For each $g\in G$, we create a map $f_g:G'/H'\rightarrow G/H: aH'\rightarrow gaH$. It's easy to see that the map is well defined a …

Question 2': We choose $G=P_{G'/H'\cup A},H=P_{\{(G'/H')-\{eH'\})\cup A}$, use natural acting of $G'$ on $G'/H'$, we can view $G'$ as subgroup of $G$ and $G/H=G'/H'\cup A$. We have the stabilizer subg …

Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$. Conjecture: There exists $k\in [n]$ such that: $$\sum_{k\in A,A\in \mathcal F} …