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 Ramsey number $R(H,\mathbb Z_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of $\mathbb Z_k$ contains a copy of $H$ whose edge labels sum to $0$ in $\mathbb Z_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over $\mathbb Z_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,\mathbb Z_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,\mathbb Z_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.
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 preferred to avoid overlooking potential violations. This paper proposes a probabilistic disaggregation framework based on Conformal Prediction (CP), a cutting-edge uncertainty quantification methodology. This framework trains a deterministic regressor to estimate normalized PV generation profiles and proposes an efficient capacity estimation algorithm to help compute the full PV generation profiles. To obtain probabilistic results, the framework applied CP with different variants, such as Mondrian Binning (MB) and Conformal Predictive System (CPS), to enhance the reliability of prediction intervals. To address the arbitrary bin count in CP with MB, the paper proposes a novel CP variant, namely: Adaptive Mondrian Binning (AMB). Its performance, along with other CP methods, is evaluated and benchmarked against quantile regression methods on two actual datasets from the region of Amsterdam, the Netherlands, and Sydney, Australia. Results show that using LightGBM as the deterministic regressor, AMB outperforms quantile regression and other CP variants. The generalizability of the proposed framework is analysed for both probabilistic outputs and key sub-processes, such as deterministic disaggregation and capacity estimation.
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 $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
Cheng Chi, Jia-Lin He, Fuhong Ma· 0 citations
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