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Edge coloring complete uniform hypergraphs with many components

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Yair Caro, Raphael Yuster

Combinatorics · Submitted 22 Feb 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$.

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arXiv IDmath/0202231 (version 1)
Pith ID6PLP5TZL, Pith's permanent record of this paper and its review
CitationRaphael Yuster, Yair Caro. "Edge coloring complete uniform hypergraphs with many components." arXiv:math/0202231 (2002). https://pith.science/paper/6PLP5TZL
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,056 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.40 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. Let $f_k(H)$ be the maximum possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. If $H$ is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. Several results concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform hypergraph on $n$ vertices, are also established.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $H$ be a hypergraph. background 1.00 0.04 0.01 unclear 0.66
s1 For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. background 0.97 0.14 0.02 asserted 0.72
s2 Let $f_k(H)$ be the maximum possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. background 0.98 0.22 0.03 asserted 0.77
s3 If $H$ is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. background 0.94 0.20 0.02 asserted 1.00
s4 In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. result 0.81 0.96 0.82 asserted 1.00
s5 Several results concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform hypergraph on $n$ vertices, are also established. contribution 0.73 0.94 0.08 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. Let $f_k(H)$ be the maximum possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. If $H$ is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. Several results concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform hypergraph on $n$ vertices, are also established.

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