11 problems

For a graph $G$ on $n$ vertices, write $e(G)$ for its number of edges and set … For nonempty vertex subsets $X,Y\subseteq V(G)$, let … Let $\sigma_{2}(G)$ denote the second-largest…

Let $r>1$, let $F$ be a graph with $e(F)>1$, let $m_1,\dots,m_r$ and $\alpha_1,\dots,\alpha_r$ be the parameters defining the symmetric multipartition property…

Let $F$ be a graph, let $m_1,\dots,m_r$ and $\alpha_1,\dots,\alpha_r$ be the parameters defining $\widetilde{\mathcal P}_{m_1,\dots,m_r}(F;\alpha_1,\dots,\alpha_r)$, and let…

Let $F$ be a graph with $m=|V(F)|$ and $e(F)>1$. For $A\subseteq V(F)$, let $e_F(A)$ be the number of edges induced by $A$, let $e_F(A^{\mathsf c})$ be the number induced by its co…

Let $F$ be a graph with $e(F)>1$, let $m_1,\dots,m_r$ be positive integers, let $\mathcal K$ be the exceptional index set, and let $\alpha_1,\dots,\alpha_r$ be the corresponding pa…

Let $r\geqslant3$, let $m_1,\dots,m_r$ be positive integers with $m=m_1+\cdots+m_r$, and let $\alpha_1,\dots,\alpha_r$ be the associated parameters. An integer $k$ is called bad wh…

Let $F$ be a graph with $e(F)>1$, let $m\geqslant 2$, and let $\widetilde{\mathcal P}_{1,\dots,1}(F;1/m,\dots,1/m)$ denote the symmetric subgraph-count property for $m$ equal parts…

Let $m$ be a positive integer, and consider subspaces of $L^2([0,1]^m)$ that are invariant under all measure-preserving bijections of $[0,1]$ onto itself, including subspaces consi…

Three-variable kernel conjecture. If $\alpha\notin\{1/3,2/3\}$, then $f=0$ almost everywhere. If $\alpha=1/3$, then, almost everywhere,

Let a generalized random or generalized quasi-random graph have $k\geq 2$ underlying clusters. Its normalized Laplacian spectrum has $k$ eigenvalues, including the zero eigenvalue,…

Induced-forcing conjecture. The path $P_3$ and its complement $\overline{P_3}$ are the only graphs not in $\mathsf{HI}$.