Generalized Turán results for intersecting cliques

Gerbner, Dániel; Patkós, Balázs

doi:10.48550/arxiv.2105.07297

Cited by 3 publications

(3 citation statements)

References 21 publications

(27 reference statements)

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“…Gerbner [8] extended this to any K a,a with a > 2 in place of C 4 . Let K + a,b denote the graph we obtain from K a,b by adding an edge to the part of size a. Gerbner and Patkós [12] showed that ex(n, K a,b , F 2 ) = N (H, K + m,n−m ) for some m. Perhaps the most natural questions concerning generalized Turán problems for double stars are counting cliques or counting double stars when a double star is forbidden. We resolve both questions by the following results.…”

Section: Introductionmentioning

confidence: 99%

Generalized Turán problems for even cycles

Gerbner

Győri

Methuku

et al. 2020

Journal of Combinatorial Theory, Series B

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Given a graph H and a set of graphs F , let ex(n, H, F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n, H, F), when H and members of F are cycles. Let C k denote the cycle of length k and let C k = {C 3 , C 4 ,. .. , C k }. We highlight the main results below. (i) We show that ex(n, C 2l , C 2k) = Θ(n l) for any l, k ≥ 2. Moreover, in some cases we determine it asymptotically: We show that ex(n, C 4 , C 2k) = (1 + o(1)) (k−1)(k−2) 4 n 2 and that the maximum possible number of C 6 's in a C 8-free bipartite graph is n 3 + O(n 5/2). (ii) Erdős's Girth Conjecture states that for any positive integer k, there exist a constant c > 0 depending only on k, and a family of graphs {G n } such that |V (G n)|= n, |E(G n)|≥ cn 1+1/k with girth more than 2k. Solymosi and Wong [38] proved that if this conjecture holds, then for any l ≥ 3 we have ex(n, C 2l , C 2l−1) = Θ(n 2l/(l−1)). We prove that their result is sharp in the sense that forbidding any other even cycle decreases the number of C 2l 's significantly: For any k > l, we have ex(n, C 2l , C 2l−1 ∪ {C 2k }) = Θ(n 2). More generally, we show that for any k > l and m ≥ 2 such that 2k = ml, we have ex(n, C ml , C 2l−1 ∪ {C 2k }) = Θ(n m). (iii) We prove ex(n, C 2l+1 , C 2l) = Θ(n 2+1/l), provided a stronger version of Erdős's Girth Conjecture holds (which is known to be true when l = 2, 3, 5). This result is also sharp in the sense that forbidding one more cycle decreases the number of C 2l+1 's significantly: More precisely, we have ex(n, C 2l+1 , C 2l ∪ {C 2k }) = O(n 2− 1 l+1), and ex(n, C 2l+1 , C 2l ∪ {C 2k+1 }) = O(n 2) for l > k ≥ 2. (iv) We also study the maximum number of paths of given length in a C k-free graph, and prove asymptotically sharp bounds in some cases.

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Section: Introductionmentioning

confidence: 99%

Generalized Turán problems for even cycles

Gerbner

Győri

Methuku

et al. 2020

Journal of Combinatorial Theory, Series B

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“…Wang [19] determined ex(n, K s , tK 2 ) for every n, s, t. Liu and Wang [13] determined ex(n, K r , 2K r ) for sufficiently large n, while Gerbner and Patkós [9] determined ex(n, K s , 2K r ) for sufficiently large n. Yuan and Yang [18] obtained a threshold on n, and determined ex(n, K 3 , 2K 3 ) for every n. A result of Gerbner [7] implies an exact result on ex(n, K s , tK r ) in the case s < r for sufficiently large. We do not state these results, as our results generalize each of them, in the case n is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Generalized Turán results for disjoint cliques

Gerbner¹

2023

Preprint

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The generalized Turán number ex(n, H, F ) is the largest number of copies of H in nvertex F -free graphs. We denote by tF the vertex-disjoint union of t copies of F . Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of ex(n, K s , tK r ). We extend this result in three directions. First, we determine ex(n, K s , tK r ) exactly for sufficiently large n. Second, we determine the asymptotics of the analogous number for p-uniform hypergraphs. Third, we determine the order of magnitude of ex(n, H, tK r ) for every graph H, and also of the analogous number for p-uniform hypergraphs.

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“…A new proof of ex(n, K r , 2K r ) can be found in [21] by Yuan and Yang. Gerbner and Patkós [9] determined ex(n, K s , 2K r ) for all s ≥ r ≥ 3 and n sufficiently large. In this paper, we determine the value of ex(n, K r , (k + 1)K r ) for all r ≥ 2, k ≥ 1 and n sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Introduction to the Department of Cardiology in Nanjing First Hospital of Nanjing Medical University, China

Zhang

Chen

2020

European Heart Journal

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Reclosure of ruptured incision after peroral endoscopic myotomy using endoloops and metallic clipsSince Inoue et al. introduced peroral endoscopic myotomy (POEM) into a clinic to treat esophageal achalasia in 2010, the procedure has been carried out in many countries around the world. 1 As more and more POEM is being done, associated technical difficulties and complications may occur. 2 To our knowledge, the present study is the first to report incision rupture after POEM.A 37-year-old man presented to our academic center after experiencing 25 years of dysphagia and 2 months of exacerbation. Barium swallow examination and esophageal manometry diagnosed the patient with type I esophageal achalasia and he agreed to receive POEM.POEM was carried out using the standard technique. After the operation, the patient received routine postoperative care. On the third day after the procedure, the patient had a fever (38.9°C, white blood cell count 18.24 × 10 9 /L, % neutrophils 95.3%) and X-ray showed the absence of several metal clips at the proximal end of the longitudinal incision which revealed the incision rupture.Gastroesophageal endoscopy showed that the middle and proximal parts of the incision was ruptured. It was not possible to reclose the incision with routine clips because of the swollen mucosa around the defect. On endoscopy, an endoloop was inserted and snared the remaining clips in the distal part. In the middle, four clips were anchored onto the defect margins at full thickness and another endoloop was inserted to snare the clips tightly. The same procedure was done in the proximal part (Figs 1,2). After monitoring the patient's condition for several days, he was discharged without any complaints or complications.In the present study, we propose reclosure of a mucosal incision after POEM using conventional endoloops and hemostatic clips. It could reclose the incision regardless of the swollen tissue or the size of the longitudinal incision.

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Generalized Turán results for intersecting cliques

Cited by 3 publications

References 21 publications

Generalized Turán problems for even cycles

Generalized Turán problems for even cycles

Generalized Turán results for disjoint cliques

Introduction to the Department of Cardiology in Nanjing First Hospital of Nanjing Medical University, China

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