5,128 research outputs found

    Monochromatic loose paths in multicolored kk-uniform cliques

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    For integers kβ‰₯2k\ge 2 and β„“β‰₯0\ell\ge 0, a kk-uniform hypergraph is called a loose path of length β„“\ell, and denoted by Pβ„“(k)P_\ell^{(k)}, if it consists of β„“\ell edges e1,…,eβ„“e_1,\dots,e_\ell such that ∣ei∩ej∣=1|e_i\cap e_j|=1 if ∣iβˆ’j∣=1|i-j|=1 and ei∩ej=βˆ…e_i\cap e_j=\emptyset if ∣iβˆ’j∣β‰₯2|i-j|\ge2. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let R(Pβ„“(k);r)R(P_\ell^{(k)};r) be the minimum integer nn such that every rr-edge-coloring of the complete kk-uniform hypergraph Kn(k)K_n^{(k)} yields a monochromatic copy of Pβ„“(k)P_\ell^{(k)}. In this paper we are mostly interested in constructive upper bounds on R(Pβ„“(k);r)R(P_\ell^{(k)};r), meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of Pβ„“(k)P_\ell^{(k)} in every coloring. In particular, we show that there is a constant c>0c>0 such that for all kβ‰₯2k\ge 2, β„“β‰₯3\ell\ge3, 2≀r≀kβˆ’12\le r\le k-1, and nβ‰₯k(β„“+1)r(1+ln⁑(r))n\ge k(\ell+1)r(1+\ln(r)), there is an algorithm such that for every rr-edge-coloring of the edges of Kn(k)K_n^{(k)}, it finds a monochromatic copy of Pβ„“(k)P_\ell^{(k)} in time at most cnkcn^k. We also prove a non-constructive upper bound R(Pβ„“(k);r)≀(kβˆ’1)β„“rR(P_\ell^{(k)};r)\le(k-1)\ell r

    Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive

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    The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W(k,2) are enormous. Much effort was put into developing lower bounds on W(k,2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lov\'asz Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W(k,2) in this light. We show how known nonconstructive lower bound proofs based on the Lov\'asz Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms

    The Wonder of Colors and the Principle of Ariadne

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    The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne Game. Some relations to other alternative. set-theoretical principles are also briefly discussed
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