1,666 research outputs found

    Ramsey numbers for partially-ordered sets

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    We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

    Diamond-free Families

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    Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of [n]:={1,...,n}[n]:=\{1,...,n\} that contains no subposet P. This problem has been studied intensively in recent years, and it is conjectured that π(P):=limnLa(n,P)/nchoosen/2\pi(P):= \lim_{n\rightarrow\infty} La(n,P)/{n choose n/2} exists for general posets P, and, moreover, it is an integer. For k2k\ge2 let \D_k denote the kk-diamond poset {A<B1,...,Bk<C}\{A< B_1,...,B_k < C\}. We study the average number of times a random full chain meets a PP-free family, called the Lubell function, and use it for P=\D_k to determine \pi(\D_k) for infinitely many values kk. A stubborn open problem is to show that \pi(\D_2)=2; here we make progress by proving \pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 page

    Universality of Entanglement and Quantum Computation Complexity

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    We study the universality of scaling of entanglement in Shor's factoring algorithm and in adiabatic quantum algorithms across a quantum phase transition for both the NP-complete Exact Cover problem as well as the Grover's problem. The analytic result for Shor's algorithm shows a linear scaling of the entropy in terms of the number of qubits, therefore difficulting the possibility of an efficient classical simulation protocol. A similar result is obtained numerically for the quantum adiabatic evolution Exact Cover algorithm, which also shows universality of the quantum phase transition the system evolves nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains a bounded quantity even at the critical point. A classification of scaling of entanglement appears as a natural grading of the computational complexity of simulating quantum phase transitions.Comment: 30 pages, 17 figures, accepted for publication in PR

    Colouring set families without monochromatic k-chains

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    A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which nn-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to kk-chain-free families. Given a family F\mathcal{F} of subsets of [n][n], we define an (r,k)(r,k)-colouring of F\mathcal{F} to be an rr-colouring of the sets without any monochromatic kk-chains F1F2FkF_1 \subset F_2 \subset \dots \subset F_k. We prove that for nn sufficiently large in terms of kk, the largest kk-chain-free families also maximise the number of (2,k)(2,k)-colourings. We also show that the middle level, ([n]n/2)\binom{[n]}{\lfloor n/2 \rfloor}, maximises the number of (3,2)(3,2)-colourings, and give asymptotic results on the maximum possible number of (r,k)(r,k)-colourings whenever r(k1)r(k-1) is divisible by three.Comment: 30 pages, final versio

    Adiabatic quantum computation and quantum phase transitions

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    We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR
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