Generalized Tur\'an problems for even cycles

Given a graph $H$ and a set of graphs $\mathcal F$, let $ex(n,H,\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal F)$, when $H$ and members of $\mathcal F$ are cycles. Let $C_k$ denote the cycle of length $k$ and let $\mathscr C_k=\{C_3,C_4,\ldots,C_k\}$. Some of our main results are the following. (i) We show that $ex(n, C_{2l}, C_{2k}) = \Theta(n^l)$ for any $l, k \ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k}) = (1+o(1)) \frac{(k-1)(k-2)}{4} n^2$ 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) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds, then for any $l \ge 3$ we have $ex(n,C_{2l},\mathscr C_{2l-1})=\Theta(n^{2l/(l-1)})$. We prove that forbidding any other even cycle decreases the number of $C_{2l}$'s significantly: For any $k > l$, we have $ex(n,C_{2l},\mathscr C_{2l-1} \cup \{C_{2k}\})=\Theta(n^2).$ More generally, we show that for any $k > l$ and $m \ge 2$ such that $2k \neq ml$, we have $ex(n,C_{ml},\mathscr C_{2l-1} \cup \{C_{2k}\})=\Theta(n^m).$ (iii) We prove $ex(n,C_{2l+1},\mathscr C_{2l})=\Theta(n^{2+1/l}),$ provided a strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true when $l = 2, 3, 5$). Moreover, forbidding one more cycle decreases the number of $C_{2l+1}$'s significantly: More precisely, we have $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k}\}) = O(n^{2-\frac{1}{l+1}}),$ and $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k+1}\}) = O(n^2)$ for $l > k \ge 2$. (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.Comment: 37 Pages; Substantially revised, contains several new results. Mistakes corrected based on the suggestions of a refere

Generalized Turán problems for even cycles

Given a graph $H$ and a set of graphs $\mathcal{F}$, let $ex(n,H,\mathcal{F})$ denote the maximum possible number of copies of $H$ in an $\mathcal{F}$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal{F})$, when $H$ and members of $\mathcal{F}$ are cycles. Let $Ck$ denote the cycle of length $k$ and let $\mathscr{C}k= C3,C4,łdots,Ck$. We highlight the main results below. (i) We show that $ex(n, C{2l}, C{2k}) = Θ(nl)$ for any $l, k \ge 2$. Moreover, in some cases we determine it asymptotically. (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 $Gn$ such that $|V(Gn)|=n$, $|E(Gn)|\ge cn{1+1/k}$ with girth more than $2k$. Solymosi and Wong proved that if this conjecture holds, then for any $l \ge 3$ we have $ex(n,C{2l},\mathscr{C}{2l-1})=Θ(n{2l/(l-1)})$. We prove that their result is sharp in the sense that forbidding any other even cycle decreases the number of $C{2l}$'s significantly. (iii) We prove $ex(n,C{2l+1},\mathscr{C}{2l})=Θ(n{2+1/l}),$ 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

Heavy metal dispersion detected in soils and plants alongside roads in Hungary

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Half-graphs, other non-stable degree sequences, and the switch Markov chain

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called $P$-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not $P$-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.Comment: Generalized the main theorem, paper extended with a number of corroborating result

The Phosphoinositide 3-Kinase Isoform PI3Kβ Regulates Osteoclast-Mediated Bone Resorption in Humans and Mice

OBJECTIVE: While phosphoinositide 3-kinases (PI3Ks) are involved in various intracellular signal transduction processes, the specific functions of the different PI3K isoforms are poorly understood. We have previously shown that the PI3Kβ isoform is required for arthritis development in the K/BxN serum–transfer model. Since osteoclasts play a critical role in pathologic bone loss during inflammatory arthritis and other diseases, we undertook this study to test the role of PI3Kβ in osteoclast development and function using a combined genetic and pharmacologic approach. METHODS: The role of PI3Kβ in primary human and murine osteoclast cultures was tested with the PI3Kβ-selective inhibitor TGX221 and by using PI3Kβ(−/−) mice. The trabecular bone architecture of PI3Kβ(−/−) mice was evaluated using micro–computed tomography and histomorphometric analyses. RESULTS: The expression of PI3Kβ was strongly and specifically up-regulated during in vitro osteoclast differentiation. In vitro development of large multinucleated osteoclasts from human or murine progenitors and their resorption capacity were strongly reduced by the PI3Kβ inhibitor TGX221 or by the genetic deficiency of PI3Kβ. This was likely due to defective cytoskeletal reorganization and vesicular trafficking, since PI3Kβ(−/−) mouse multinucleated cells failed to form actin rings and retained intracellular acidic vesicles and cathepsin K. In contrast, osteoclast-specific gene expression and the survival and apoptosis of osteoclasts were not affected. PI3Kβ(−/−) mice had significantly increased trabecular bone volume and showed abnormal osteoclast morphology with defective resorption pit formation. CONCLUSION: PI3Kβ plays an important role in osteoclast development and function and is required for in vivo bone homeostasis

Diszkrét matematika = Discrete mathematics

A pályázat résztvevői igen aktívak voltak a 2006-2008 években. Nemcsak sok eredményt értek el, miket több mint 150 cikkben publikáltak, eredményesen népszerűsítették azokat. Több mint 100 konferencián vettek részt és adtak elő, felerészben meghívott, vagy plenáris előadóként. Hagyományos gráfelmélet Több extremális gráfproblémát oldottunk meg. Új eredményeket kaptunk Ramsey számokról, globális és lokális kromatikus számokról, Hamiltonkörök létezéséséről. a crossig numberről, gráf kapacitásokról és kizárt részgráfokról. Véletlen gráfok, nagy gráfok, regularitási lemma Nagy gráfok "hasonlóságait" vizsgáltuk. Különféle metrikák ekvivalensek. Űj eredeményeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. Hipergráfok, egyéb kombinatorika Új Sperner tipusú tételekte kaptunk, aszimptotikusan meghatározva a halmazok max számát bizonyos kizárt struktőrák esetén. Több esetre megoldottuk a kizárt hipergráf problémát is. Elméleti számítástudomány Új ujjlenyomat kódokat és bioinformatikai eredményeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The "similarities" of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found

Hematopoietic or osteoclast-specific deletion of Syk leads to increased bone mass in experimental mice

<p>Syk is a non-receptor tyrosine kinase critically involved in signaling by various immunoreceptors including B-cell-receptors and activating Fc-receptors. We have previously shown that Syk also mediates immunoreceptor-like signals required for the in vitro development and function of osteoclasts. However, the perinatal lethality of Syk^−/− mice precluded the analysis of the role of Syk in in vivo bone metabolism. To overcome that problem, we generated mice with osteoclast-specific (Syk^ΔOC) or hematopoietic (Syk^ΔHaemo) Syk deficiency by conditional deletion of Syk using Cre recombinase expressed under the control of the Ctsk or Vav1 promoter, respectively. Micro-CT analysis revealed increased bone trabecular density in both Syk^ΔOC and Syk^ΔHaemo mice, although hematopoietic Syk deficiency caused a more severe phenotype than osteoclast-specific Syk deficiency. Osteoclast-specific Syk deficiency reduced, whereas hematopoietic Syk deficiency completely blocked in vitro development of osteoclasts. Both interventions inhibited the resorptive activity of osteoclasts and osteoclast-specific gene expression. Kinetic analysis of Syk protein levels, Cre expression and the genomic deletion of the Syk^flox allele revealed complete and early deletion of Syk from Syk^ΔHaemo osteoclasts whereas Syk was incompletely deleted at a later stage of osteoclast development from Syk^ΔOC cultures. Those results provide an explanation for the in vivo and in vitro difference between the Syk^ΔOC and Syk^ΔHaemo mutant strains and suggest late activation of, and incomplete target gene deletion upon, osteoclast-specific Cre expression driven by the Ctsk promoter. Taken together, our results indicate that Syk plays an indispensable role in osteoclast-mediated in vivo bone resorption and suggest that Syk-specific inhibitors may provide therapeutic benefit in inflammatory and other diseases characterized by excessive osteoclast-mediated bone resorption.</p