Ryser Type Conditions for Extending Colorings of Triples

In 1951, Ryser showed that an $n\times n$ array $L$ whose top left $r\times s$ subarray is filled with $n$ different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order $n$ if and only if the number of occurrences of each symbol in $L$ is at least $r+s-n$. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let $X,Y$ be finite sets with $X\subsetneq Y$ and $|Y|\equiv 0 \pmod 3$. When can we extend a (proper) coloring of $\lambda \binom{X}{3}$ (all triples on a ground set $X$, each one being repeated $\lambda$ times) to a coloring of $\lambda \binom{Y}{3}$ using the fewest possible number of colors? It is necessary that the number of triples of each color in $\binom{X}{3}$ is at least $|X|-2|Y|/3$. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483--495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using H\"aggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295--313), we show that the number of triples of each color being at least $|X|/2-|Y|/6$ is sufficient. Finally we prove an Evans type result by showing that if $|Y|\geq 3|X|$, then any $q$-coloring of any subset of $\lambda \binom{X}{3}$ can be embedded into a $\lambda\binom{|Y|-1}{2}$-coloring of $\lambda \binom{Y}{3}$ as long as $q\leq \lambda \binom{|Y|-1}{2}-\lambda \binom{|X|}{3}/\lfloor{|X|/3}\rfloor$.Comment: 10 page

Social inclusion of vulnerable groups through participatory and emancipatory approaches. Implementing active citizenship and socially innovative actions in the framework of civil & human rights model of disability

The research hypothesis of the thesis is that “an open participation in the co-creation of the services and environments, makes life easier for vulnerable groups”; assuming that the participatory and emancipatory approaches are processes of possible actions and changes aimed at facilitating people’s lives. The adoption of these approaches is put forward as the common denominator of social innovative practices that supporting inclusive processes allow a shift from a medical model to a civil and human rights approach to disability. The theoretical basis of this assumption finds support in many principles of Inclusive Education and the main focus of the hypothesis of research is on participation and emancipation as approaches aimed at facing emerging and existing problems related to inclusion. The framework of reference for the research is represented by the perspectives adopted by several international documents concerning policies and interventions to promote and support the leadership and participation of vulnerable groups. In the first part an in-depth analysis of the main academic publications on the central themes of the thesis has been carried out. After investigating the framework of reference, the analysis focuses on the main tools of participatory and emancipatory approaches, which are able to connect with the concepts of active citizenship and social innovation. In the second part two case studies concerning participatory and emancipatory approaches in the areas of concern are presented and analyzed as example of the improvement of inclusion, through the involvement and participation of persons with disability. The research has been developed using a holistic and interdisciplinary approach, aimed at providing a knowledge-base that fosters a shift from a situation of passivity and care towards a new scenario based on the person’s commitment in the elaboration of his/her own project of life