11,848 research outputs found

    Integer colorings with forbidden rainbow sums

    Full text link
    For a set of positive integers A⊆[n]A \subseteq [n], an rr-coloring of AA is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of [n][n] with the maximum number of rainbow sum-free rr-colorings. We show that for r=3r=3, the interval [n][n] is optimal, while for r≥8r\geq8, the set [⌊n/2⌋,n][\lfloor n/2 \rfloor, n] is optimal. We also prove a stability theorem for r≥4r\geq4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

    Some local--global phenomena in locally finite graphs

    Full text link
    In this paper we present some results for a connected infinite graph GG with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of GG. (For a vertex ww of a graph GG the ball of radius rr centered at ww is the subgraph of GG induced by the set Mr(w)M_r(w) of vertices whose distance from ww does not exceed rr). In particular, we prove that if every ball of radius 2 in GG is 2-connected and GG satisfies the condition dG(u)+dG(v)≥∣M2(w)∣−1d_G(u)+d_G(v)\geq |M_2(w)|-1 for each path uwvuwv in GG, where uu and vv are non-adjacent vertices, then GG has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in GG satisfies Ore's condition (1960) then all balls of any radius in GG are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

    Automated Discharging Arguments for Density Problems in Grids

    Full text link
    Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of 37≈0.428571\frac{3}{7} \approx 0.428571 and a lower bound of 512≈0.416666\frac{5}{12}\approx 0.416666. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of 2355≈0.418181\frac{23}{55} \approx 0.418181 on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure

    Complex Hadamard matrices and Equiangular Tight Frames

    Full text link
    In this paper we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. and extend the list of known equiangular tight frames. In particular, a (36,21) frame coming from a nontrivial cube root signature matrix is obtained for the first time.Comment: 6 pages, contribution to the 16th ILAS conference, Pisa, 201
    • …
    corecore