237 research outputs found

    An extension of Tur\'an's Theorem, uniqueness and stability

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    We determine the maximum number of edges of an nn-vertex graph GG with the property that none of its rr-cliques intersects a fixed set M⊂V(G)M\subset V(G). For (r−1)∣M∣≥n(r-1)|M|\ge n, the (r−1)(r-1)-partite Turan graph turns out to be the unique extremal graph. For (r−1)∣M∣<n(r-1)|M|<n, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's comments incorporate

    Edge-decompositions of graphs with high minimum degree

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    A fundamental theorem of Wilson states that, for every graph FF, every sufficiently large FF-divisible clique has an FF-decomposition. Here a graph GG is FF-divisible if e(F)e(F) divides e(G)e(G) and the greatest common divisor of the degrees of FF divides the greatest common divisor of the degrees of GG, and GG has an FF-decomposition if the edges of GG can be covered by edge-disjoint copies of FF. We extend this result to graphs GG which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3K_3-divisible graph of minimum degree at least 9n/10+o(n)9n/10+o(n) has a K3K_3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3K_3-divisible graph with minimum degree at least 3n/43n/4 has a K3K_3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3+o(n)2n/3 +o(n) for the existence of a C4C_4-decomposition, and of n/2+o(n)n/2+o(n) for the existence of a C2ℓC_{2\ell}-decomposition, where ℓ≥3\ell\ge 3. Our main contribution is a general `iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3K_3-divisible graph with minimum degree at least 3n/4+o(n)3n/4+o(n) has an approximate K3K_3-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement

    Semi-algebraic Ramsey numbers

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    Given a finite point set P⊂RdP \subset \mathbb{R}^d, a kk-ary semi-algebraic relation EE on PP is the set of kk-tuples of points in PP, which is determined by a finite number of polynomial equations and inequalities in kdkd real variables. The description complexity of such a relation is at most tt if the number of polynomials and their degrees are all bounded by tt. The Ramsey number Rkd,t(s,n)R^{d,t}_k(s,n) is the minimum NN such that any NN-element point set PP in Rd\mathbb{R}^d equipped with a kk-ary semi-algebraic relation EE, such that EE has complexity at most tt, contains ss members such that every kk-tuple induced by them is in EE, or nn members such that every kk-tuple induced by them is not in EE. We give a new upper bound for Rkd,t(s,n)R^{d,t}_k(s,n) for k≥3k\geq 3 and ss fixed. In particular, we show that for fixed integers d,t,sd,t,s, R3d,t(s,n)≤2no(1),R^{d,t}_3(s,n) \leq 2^{n^{o(1)}}, establishing a subexponential upper bound on R3d,t(s,n)R^{d,t}_3(s,n). This improves the previous bound of 2nC2^{n^C} due to Conlon, Fox, Pach, Sudakov, and Suk, where CC is a very large constant depending on d,t,d,t, and ss. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in Rd\mathbb{R}^d. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

    An extension of Turán's theorem, uniqueness and stability

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    We determine the maximum number of edges of an n -vertex graph G with the property that none of its r -cliques intersects a fixed set M⊂V(G) . For (r−1)|M|≥n , the (r−1) -partite Turán graph turns out to be the unique extremal graph. For (r−1)|M|<n , there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results
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