Generalized cohesiveness

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set $A$ of natural numbers is $n$--cohesive (respectively, $n$--r--cohesive) if $A$ is almost homogeneous for every computably enumerable (respectively, computable) $2$--coloring of the $n$--element sets of natural numbers. (Thus the $1$--cohesive and $1$--r--cohesive sets coincide with the cohesive and r--cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of $n$--cohesive and $n$--r--cohesive sets. For example, we show that for all $n \ge 2$, there exists a $\Delta^0_{n+1}$ $n$--cohesive set. We improve this result for $n = 2$ by showing that there is a $\Pi^0_2$ $2$--cohesive set. We show that the $n$--cohesive and $n$--r--cohesive degrees together form a linear, non--collapsing hierarchy of degrees for $n \geq 2$. In addition, for $n \geq 2$ we characterize the jumps of $n$--cohesive degrees as exactly the degrees {\bf \geq \jump{0}{(n+1)}} and show that each $n$--r--cohesive degree has jump {\bf > \jump{0}{(n)}}

Ramsey-type graph coloring and diagonal non-computability

A function is diagonally non-computable (d.n.c.) if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over omega-models.Comment: 18 page

Understanding the heterogeneity of the hematopoietic stem cells

The hematopoietic system is replenished and maintained throughout life by rare hematopoietic stem cells (HSCs) that reside in the bone marrow (BM) of adult mammals. Over the last 20 years, the advancement in the field lead to the acknowledgement of the heterogeneity within the HSC compartment unraveling the presence of HSC subsets with certain mature blood lineage preferences so called lineage-biased (Li-bi) HSCs. Studying the heterogeneity and lineage bias within the HSC compartment is crucial not only to understand the functional and molecular mechanisms behind this lineage skewing but can also shed light on the emergence of hematological malignancies subsequently paving the way to find new therapeutic targets, better treatment options and more selective alternatives of BM transplantation. Recent developments have taken advantage of immunophenotypic markers for prospective isolation of cells. The cell surface markers can be used to enrich for HSCs but cannot purify. Current markers cannot resolve heterogeneity within the HSC compartment, highlighting the importance of continuing efforts on identifying new cell surface markers that enrich Li-bi HSC subtypes. In paper I, we demonstrate that CD49b cell surface marker subfractionates the most primitive HSC compartment into two; CD49b– HSCs with myeloid bias, high self-renewal potential and the most quiescent state, and CD49b+ HSCs with lymphoid bias, lowered selfrenewal potential and more proliferative state. Furthermore, we show that both subsets have similar transcriptome profiles but distinct epigenetic landscapes highlighting that the lineage-bias is regulated via epigenetic mechanisms. In paper III, we show that using the additional cell surface marker CD229, the remaining heterogeneity within the CD49b+ HSCs can be resolved into two functional subsets as CD49b+CD229– and CD49b+CD229+. The CD49b+CD229– fraction shows long-term and stable reconstitution and the CD49b+CD229+ fraction enriches for multipotent progenitor cells having short term activity. Hematopoietic aging is associated with myeloid skewing, delayed, and reduced immune response and higher incidences of myeloid malignancies. The composition of HSC compartment changes with a shift toward an increased proportion of myeloid biased HSCs in elderly both in human and mice. However, the molecular mechanisms behind this phenomenon are not completely understood. In paper II, we show that the CD49b– HSC maintains its myeloid bias in the peripheral blood of the young, young adult and old age groups whereas the CD49b+ HSC shifts from lymphoid bias in young and young adult to lineage-balance (no bias) in aged mice. In addition, we demonstrate that both subsets are equally active in young and have similar chromatin landscapes with different levels of accessible regions in old mice. The B cell lineage priming occurs downstream of HSCs starting at the branching point of multipotent progenitors in the hematopoietic hierarchy. The B cell development is highly regulated by transcriptional factors. In paper IV, we show that combined loss of transcription factors FOXO1 and FOXO3 prevents the B cell development by blocking it at the BLP stage. Moreover, we demonstrate that FOXO3 plays a crucial role in regulating the B cell lineage priming higher up in the hematopoietic hierarchy already as early as the LMPP level. Collectively, this thesis identifies cell surface markers that resolves the functional heterogeneity of the HSCs, gives insights into how the lineage bias is regulated during aging, and unravels the effect of transcription factors in B cell development