908 research outputs found

    Ramsey properties of randomly perturbed graphs: cliques and cycles

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    Given graphs H1,H2H_1,H_2, a graph GG is (H1,H2)(H_1,H_2)-Ramsey if for every colouring of the edges of GG with red and blue, there is a red copy of H1H_1 or a blue copy of H2H_2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3,Kt)(K_3,K_t)-Ramsey (for tβ‰₯3t\ge 3). They also raised the question of generalising this result to pairs of graphs other than (K3,Kt)(K_3,K_t). We make significant progress on this question, giving a precise solution in the case when H1=KsH_1=K_s and H2=KtH_2=K_t where s,tβ‰₯5s,t \ge 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3,Kt)(K_3,K_t)-Ramsey question. Moreover, we give bounds for the corresponding (K4,Kt)(K_4,K_t)-Ramsey question; together with a construction of Powierski this resolves the (K4,K4)(K_4,K_4)-Ramsey problem. We also give a precise solution to the analogous question in the case when both H1=CsH_1=C_s and H2=CtH_2=C_t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs,Kt)(C_s,K_t)-Ramsey (for odd sβ‰₯5s\ge 5 and tβ‰₯4t\ge 4).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil Powierski, stated results for cliques in graphs of low positive density separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to appear in CP

    Combinatorial theorems relative to a random set

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    We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

    Generalized List Decoding

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    This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR\textsf{XOR}) channels, erasure channels, AND\textsf{AND} (ZZ-) channels, OR\textsf{OR} channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (Lβˆ’1)(L-1)-list decodable codes (where the list size LL is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate (Lβˆ’1)(L-1)-list decodable codes if and only if the set of completely positive tensors of order-LL with admissible marginals is not entirely contained in the order-LL confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large LL for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting

    Large rainbow cliques in randomly perturbed dense graphs

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    For two graphs GG and HH, write G⟢rbwHG \stackrel{\mathrm{rbw}}{\longrightarrow} H if GG has the property that every {\sl proper} colouring of its edges yields a {\sl rainbow} copy of HH. We study the thresholds for such so-called {\sl anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form GβˆͺG(n,p)G \cup \mathbb{G}(n,p), where GG is an nn-vertex graph with edge-density at least dd, and dd is a constant that does not depend on nn. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property GβˆͺG(n,p)⟢rbwKsG \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s for every ss. In this paper, we show that for sβ‰₯9s \geq 9 the threshold is nβˆ’1/m2(K⌈s/2βŒ‰)n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}; in fact, our 11-statement is a supersaturation result. This turns out to (almost) be the threshold for s=8s=8 as well, but for every 4≀s≀74 \leq s \leq 7, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property GβˆͺG(n,p)⟢rbwC2β„“βˆ’1G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}, and show that the threshold for this property is nβˆ’2n^{-2} for every β„“β‰₯2\ell \geq 2; in particular, it does not depend on the length of the cycle C2β„“βˆ’1C_{2\ell - 1}. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.Comment: 21 pages; some typos fixed in the last versio
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