TY - JOUR

T1 - ON MULTICOLOR RAMSEY NUMBERS AND SUBSET COLORING OF HYPERGRAPHS

AU - Jartoux, Bruno

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

AU - Yuditsky, Yelena

N1 - Publisher Copyright:
© 2022 Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky.

PY - 2022

Y1 - 2022

N2 - For n ≥ s > r ≥ 1 and k ≥ 2, write n → (s)r k if every hyperedge coloring with k colors of the complete r-uniform hypergraph on n vertices has a monochromatic subset of size s. Improving upon previous results by M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi [Discrete Math., 322 (2014), pp. 69-77] and P. Erdos, A. Hajnal, A. Máté, and R. Rado, [Combinatorial set theory: Partition Relations for Cardinals, Elsevier, Amsterdam, 1984] we show that if r ≥ 3 and n → (s)r k, then 2n → (s+1)r+1 k+3. This improves some of the known lower bounds on multicolor hypergraph Ramsey numbers. Given a hypergraph H = (V,E), we consider the Ramsey-like problem of coloring all r-subsets of V such that no hyperedge of size ≥ r + 1 is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number χ(H). In particular we show that this number is O(log(r-1)(rχ (H))+r), where logy is the log function applied y times.

AB - For n ≥ s > r ≥ 1 and k ≥ 2, write n → (s)r k if every hyperedge coloring with k colors of the complete r-uniform hypergraph on n vertices has a monochromatic subset of size s. Improving upon previous results by M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi [Discrete Math., 322 (2014), pp. 69-77] and P. Erdos, A. Hajnal, A. Máté, and R. Rado, [Combinatorial set theory: Partition Relations for Cardinals, Elsevier, Amsterdam, 1984] we show that if r ≥ 3 and n → (s)r k, then 2n → (s+1)r+1 k+3. This improves some of the known lower bounds on multicolor hypergraph Ramsey numbers. Given a hypergraph H = (V,E), we consider the Ramsey-like problem of coloring all r-subsets of V such that no hyperedge of size ≥ r + 1 is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number χ(H). In particular we show that this number is O(log(r-1)(rχ (H))+r), where logy is the log function applied y times.

KW - hypergraph Ramsey numbers

KW - multicolor Ramsey numbers

KW - subset coloring

UR - http://www.scopus.com/inward/record.url?scp=85137156867&partnerID=8YFLogxK

U2 - 10.1137/21M1462003

DO - 10.1137/21M1462003

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AN - SCOPUS:85137156867

SN - 0895-4801

VL - 36

SP - 1848

EP - 1860

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

ER -