10 research outputs found

    A note on lower bounds for hypergraph Ramsey numbers

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    We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that r3(l,l,l)2lcloglogl.r_3 (l,l,l) \geq 2^{l^{c \log \log l}}. The old bound, due to Erd\H{o}s and Hajnal, was r3(l,l,l)2cl2log2l.r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.Comment: 6 page

    Hypergraph Ramsey numbers of cliques versus stars

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    Let Km(3)K_m^{(3)} denote the complete 33-uniform hypergraph on mm vertices and Sn(3)S_n^{(3)} the 33-uniform hypergraph on n+1n+1 vertices consisting of all (n2)\binom{n}{2} edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number r(K4(3),Sn(3))r(K_{4}^{(3)},S_n^{(3)}) exhibits an unusual intermediate growth rate, namely, 2clog2nr(K4(3),Sn(3))2cn2/3logn 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} for some positive constants cc and cc'. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum NN such that any 22-edge-coloring of the Cartesian product KNKNK_N \square K_N contains either a red rectangle or a blue KnK_n?Comment: 13 page

    Large cliques or co-cliques in hypergraphs with forbidden order-size pairs

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    The well-known Erd\H{o}s-Hajnal conjecture states that for any graph FF, there exists ϵ>0\epsilon>0 such that every nn-vertex graph GG that contains no induced copy of FF has a homogeneous set of size at least nϵn^{\epsilon}. We consider a variant of the Erd\H{o}s-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on mm vertices and ff edges for any positive mm and 0f(m2)0\leq f \leq \binom{m}{2}, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case m=4m=4, for every S{0,1,2,3,4}S \subseteq \{0,1,2,3,4\}, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in SS. In most cases the bounds are essentially tight. We also determine, for all SS, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.Comment: A preliminary version of this manuscript appeared as arXiv:2303.0957

    Large cliques or cocliques in hypergraphs with forbidden order-size pairs

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    The well-known Erdős-Hajnal conjecture states that for any graph FF, there exists ϵ>0ϵ>0 such that every nn-vertex graph GG that contains no induced copy of FF has a homogeneous set of size at least nϵn^ϵ. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on mm vertices and ff edges for any positive mm and 0f(m2)0≤f≤(m2), then we obtain large homogeneous sets. For triple systems, in the first nontrivial case m=4m=4, for every S0,1,2,3,4S⊆{0,1,2,3,4}, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in SS. In most cases the bounds are essentially tight. We also determine, for all SS, whether the growth rate is polynomial or polylogarithmic. Some open problems remain

    Semi-algebraic and semi-linear Ramsey numbers

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    An rr-uniform hypergraph HH is semi-algebraic of complexity t=(d,D,m)\mathbf{t}=(d,D,m) if the vertices of HH correspond to points in Rd\mathbb{R}^{d}, and the edges of HH are determined by the sign-pattern of mm degree-DD polynomials. Semi-algebraic hypergraphs of bounded complexity provide a general framework for studying geometrically defined hypergraphs. The much-studied semi-algebraic Ramsey number Rrt(s,n)R_{r}^{\mathbf{t}}(s,n) denotes the smallest NN such that every rr-uniform semi-algebraic hypergraph of complexity t\mathbf{t} on NN vertices contains either a clique of size ss, or an independent set of size nn. Conlon, Fox, Pach, Sudakov, and Suk proved that R_{r}^{\mathbf{t}}(n,n)<\mbox{tw}_{r-1}(n^{O(1)}), where \mbox{tw}_{k}(x) is a tower of 2's of height kk with an xx on the top. This bound is also the best possible if min{d,D,m}\min\{d,D,m\} is sufficiently large with respect to rr. They conjectured that in the asymmetric case, we have R3t(s,n)<nO(1)R_{3}^{\mathbf{t}}(s,n)<n^{O(1)} for fixed ss. We refute this conjecture by showing that R3t(4,n)>n(logn)1/3o(1)R_{3}^{\mathbf{t}}(4,n)>n^{(\log n)^{1/3-o(1)}} for some complexity t\mathbf{t}. In addition, motivated by results of Bukh-Matou\v{s}ek and Basit-Chernikov-Starchenko-Tao-Tran, we study the complexity of the Ramsey problem when the defining polynomials are linear, that is, when D=1D=1. In particular, we prove that Rrd,1,m(n,n)2O(n4r2m2)R_{r}^{d,1,m}(n,n)\leq 2^{O(n^{4r^2m^2})}, while from below, we establish Rr1,1,1(n,n)2Ω(nr/21)R^{1,1,1}_{r}(n,n)\geq 2^{\Omega(n^{\lfloor r/2\rfloor-1})}.Comment: 23 pages, 1 figur

    THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

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    and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

    New lower bounds for hypergraph Ramsey numbers

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