For a fixed graph $T$ and a fixed nonempty family $\mathcal{F}$ of graphs, let $\operatorname{ex}(G, T, \mathcal{F})$ denote the maximum number of copies of $T$ in an $\mathcal{F}$-free subgraph of a graph $G$. Alon and Shikhelman~[Algorithmica, 2022] conjectured that if $e(T)>0$ and no member of $\mathcal{F}$ is a sub...
Yi-Han Chen, Jia-Lin He, Xi-He Li et al.· 0 citations
For a graph $G$ with $3\mid e(G)$, let $R(G,\mathbb Z_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to\mathbb Z_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge6$ and $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le n+5$. Moreover, the st...
We prove a sparse version of the four-colour theorem of Brandt and Thomass\'{e}, answering a question of Allen, B\"ottcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn...
Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\geq2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramse...
Disaggregating solar photovoltaics (PV) profiles from smart electricity meter data has attracted attention, as Distribution System Operators (DSOs) need street-level PV generation profiles to improve grid operations and planning. Given the importance of reliability in operational decisions, probabilistic results are pr...
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$....
Cheng Chi, Jia-Lin He, Fuhong Ma· 0 citations
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