The maximum number of PℓP_\ell copies in PkP_k-free graphs

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai

VAV1 NEL DIFFERENZIAMENTO MONOCITO/MACROFAGICO DI PRECURSORI MIELOIDI TUMORALI

Vav1 is a critical signal transducer for the development and function of normal hematopoietic cells, in which it regulates the acquisition of maturation-related properties, including adhesion, motility, and phagocytosis. In addition, Vav1is a key player in the ATRAinduced completion of the differentiation program of tumoral myeloid precursors derived from APL, in which it promotes the acquisition of a mature phenotype by playing multiple functions at both cytoplasmic and nuclear levels. Here we investigate the possible role of Vav1 in the differentiation of leukemic precursors to monocytes/macrophages. Tumoral promyelocytes in which Vav1was negatively modulated were induced to differentiate along the monocytic/macrophagic lineage with ATRA and PMA and monitored for their maturation-related properties. We found that Vav1 is crucial for the phenotypical differentiation of tumoral myeloid precursors to monocytes/macrophages, in terms of CD11b expression, adhesion capability and cell morphology. Confocal analysis revealed that Vav1 may synergize with actin in modulating nuclear morphology of PMA-treated adherent cells. Moreover, Electrophoretic Mobility Shift Assays indicated that Vav1 and the transcription factor PU.1 are recruited to CD11b promoter, suggesting that the two proteins cooperate to regulate the expression of the surface antigen CD11b. The reported results constitute the first evidence thatVav1 plays a crucial role in the maturation of tumoral myeloid precursors to monocytes/macrophages. Since Vav1 is also critical for the maturation of leukemic promyelocytes along the granulocytic lineage, our data highlight the key role for this protein during the completion of the differentiation program of tumoral myeloid cells along the various hematopoietic lineages and suggest that Vav1 is a common target for developing future treatment strategies for the diverse subtypes of myeloid leukemias

A note on the maximum number of triangles in a C5-free graph

We prove that the maximum number of triangles in a C5-free graph on n vertices is at most [Formula presented](1+o(1))n3/2, improving an estimate of Alon and Shikhelman [Alon, N. and C. Shikhelman, Many T copies in H-free graphs. Journal of Combinatorial Theory, Series B 121 (2016) 146-172]. © 2017 Elsevier B.V

Inverse Tur\'an numbers

For given graphs $G$ and $F$, the Tur\'an number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number $k$, one maximizes the number of edges in a host graph $G$ for which $ex(G,H) < k$. Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Tur\'an number of the paths of length $4$ and $5$ and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inverse Tur\'an number of even cycles giving improved bounds on the leading coefficient in the case of $C_4$. Finally, we give multiple conjectures concerning the asymptotic value of the inverse Tur\'an number of $C_4$ and $P_{\ell}$, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of $\ell$.Comment: updated to include the suggestions of reviewer

Subgraph densities in KrK_{r}-free graphs

In this paper we disprove a conjecture of Lidický and Murphy about the number of copies of a given graph in a $K_{r}$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most~2 in a triangle-free graph

Tur\'an numbers of Berge trees

A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, we determine the Tur\'an number of a hypergraph without a Berge copy of a tree with $k$ edges, for all $k$ and $r$, $r \ge k(k-2)$ and infinitely many $n$. We also characterize the extremal hypergraphs for these values