Abstract: Vopenka's Principle (VP) asserts that the category Ord is not fully embeddable into the category of Graphs, whereas the Weak Vopenka's Principle (WVP) asserts that the category $Ord^{op}$ is not fully embeddable into the category of Graphs. Although VP is a well-known very strong large-cardinal principle, implying a proper class of extendible cardinals, until recently it was only known that WVP implies the existence of a proper class of measurable cardinals. This contrasts with the widely believed conjecture that WVP would be, if not equivalent to VP, it would at least have similar consistency strength. The question has been answered very recently by Trevor Wilson in a totally unexpected way. He showed that WVP is in fact equivalent to the assertion that the class of ordinals (ORD) is Woodin. In this series of talks we'll begin with a presentation of Wilson's proof with certain detail. We will then work through the level-by-level versions, showing that, e.g., WVP restricted to $\Sigma_2$-definable classes of graphs is equivalent to the existence of a strong cardinal. We will also give some historical background and point to some applications.

Abstract: Lectures 1 and 2: We give a systematic introduction to some of the basic notions of combinatorial Set Theory as $\Delta$-systems, partition relations, set mappings and polarized partitions.
Lecture 3:We investigate the properties of the coloring number, a friendly replacement of chromatic number.

Abstract: Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show that $\mathcal{C}$ might have minimal elements which answers a question due to Galvin.

Abstract: 1st Talk: In 2014, the 2019 Hausdorff medal laureate I. Neeman formulated an elegant framework to iterate proper forcings with finite conditions, and give some applications, where the framework notably enabled him to reprove the consistency of PFA with finite conditions. This initiated speculation among experts that may a strong higher version of forcing axiom exist. His general approach known today as "forcing with side conditions" has found many applications since his discovery. There are two main directions one can go with this phenomenon, designing new forcing notions with this technology to answer far-reaching and hard problems which previously seemed hopeless to be settled, or searching for higher forcing axioms. Several experts found various presentations of his original idea and leaded to settle new results, among them, B. Velickovic founded a framework to iterate semi-proper forcings which is in favour of the first direction, but his nontrivial construction resulted in the birth of an impressive concept, that of virtual model which this talk is dedicated to its development, and its appearance in Velickovic's approach towards the iteration. In joint work with Velickovic, we used virtual models to establish the consistency of some combinatorial principles simultaneously which will be the subject of my second talk.

2nd Talk:C. Weiss formulated some combinatorial properties that capture the essence of some large cardinal properties, mostly at level of supercompactness, but can hold at small cardinals. He and M. Viale could then used them to show that any standard forcing construction of a model of PFA requires at least a strongly compact cardinal. In this talk, I will introduce $GM^+(\omega_3,\omega_1) $ a combinatorial principle strengthening those by Weiss, and will discuss also the consequences of it. This is joint work with B. Velickovic.

Abstract: Todorcevic introduced a method of forcing with models as side conditions, so called the side condition method. This method is a strong tool to add an uncountable set by forcing. By combining the side condition method with some preservation theorems, it is enable us to obtain new consistency results. The followings are samples of structures that are preserved by some side condition method.
1- a non-special Aronszajn tree.
2- the covering number of the Lebesgue measure zero ideal equal to $\aleph_1$.
3- an n-entangled set of reals (n is an integer not smaller than 2)
YPFA is one of weak fragments of PFA, introduced by Chodounsky-Zapletal. By use of side condition method, it is shown that YPFA implies Moor's Mapping Reflection Principle. A part of my talk is a joint work with Tadatoshi Miyamoto (Nanzan University).