93 research outputs found

    Generalized Kneser coloring theorems with combinatorial proofs

    Full text link
    The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.Comment: 19 pages, 4 figure

    Tournaments, 4-uniform hypergraphs, and an exact extremal result

    Full text link
    We consider 44-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of 55 vertices spans either 00 or exactly 22 hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a `switching' operation on tournaments that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure

    Balanced walls for random groups

    Get PDF
    We study a random group G in the Gromov density model and its Cayley complex X. For density < 5/24 we define walls in X that give rise to a nontrivial action of G on a CAT(0) cube complex. This extends a result of Ollivier and Wise, whose walls could be used only for density < 1/5. The strategy employed might be potentially extended in future to all densities < 1/4.Comment: 18 pages, 2 figures. v2: Minor improvements, final versio

    Combinatorial Stokes formulas via minimal resolutions

    Get PDF
    We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn implies "Dold's theorem" that there is no equivariant map from an n-connected to an n-dimensional free Z_k-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).Comment: 18 page

    Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

    Get PDF
    A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023)

    The codegree threshold of K4K_4^-

    Get PDF
    The codegree threshold ex2(n,F)\mathrm{ex}_2(n, F) of a 33-graph FF is the minimum d=d(n)d=d(n) such that every 33-graph on nn vertices in which every pair of vertices is contained in at least d+1d+1 edges contains a copy of FF as a subgraph. We study ex2(n,F)\mathrm{ex}_2(n, F) when F=K4F=K_4^-, the 33-graph on 44 vertices with 33 edges. Using flag algebra techniques, we prove that if nn is sufficiently large then ex2(n,K4)(n+1)/4\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration GG, there is a quasirandom tournament TT on the same vertex set such that GG is close in the edit distance to the 33-graph C(T)C(T) whose edges are the cyclically oriented triangles from TT. For infinitely many values of nn, we are further able to determine ex2(n,K4)\mathrm{ex}_2(n, K_4^-) exactly and to show that tournament-based constructions C(T)C(T) are extremal for those values of nn.Comment: 31 pages, 7 figures. Ancillary files to the submission contain the information needed to verify the flag algebra computation in Lemma 2.8. Expands on the 2017 conference paper of the same name by the same authors (Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413

    Transforming Monitoring Structures with Resilient Encoders. Application to Repeated Games

    Full text link
    An important feature of a dynamic game is its monitoring structure namely, what the players effectively see from the played actions. We consider games with arbitrary monitoring structures. One of the purposes of this paper is to know to what extent an encoder, who perfectly observes the played actions and sends a complementary public signal to the players, can establish perfect monitoring for all the players. To reach this goal, the main technical problem to be solved at the encoder is to design a source encoder which compresses the action profile in the most concise manner possible. A special feature of this encoder is that the multi-dimensional signal (namely, the action profiles) to be encoded is assumed to comprise a component whose probability distribution is not known to the encoder and the decoder has a side information (the private signals received by the players when the encoder is off). This new framework appears to be both of game-theoretical and information-theoretical interest. In particular, it is useful for designing certain types of encoders that are resilient to single deviations and provide an equilibrium utility region in the proposed setting; it provides a new type of constraints to compress an information source (i.e., a random variable). Regarding the first aspect, we apply the derived result to the repeated prisoner's dilemma.Comment: Springer, Dynamic Games and Applications, 201
    corecore