This monograph explores mathematical constructions essential to data mining, particularly in pattern recognition. It employs combinatorial and graph theoretic techniques to examine infeasible systems of linear inequalities, whose generalized solutions serve as foundational elements for geometric decision rules in pattern recognition. These infeasible systems are crucial in geometric descriptions of pattern recognition problems, facilitated by the committee method. They represent a significant subclass of infeasible constraint systems with a monotonicity property, where the multi-indices of feasible subsystems form abstract simplicial complexes—key objects in combinatorial topology. The data mining and machine learning methods presented here underpin technologies such as big data and deep learning, which increasingly influence human-technology interactions and enhance problem-solving capabilities. The contents include sections on the relationship between pattern recognition and infeasible systems, the role of complexes and graphs, the connection between polytopes and inequality systems, and the interplay of monotone Boolean functions with these concepts. Additionally, the monograph provides a bibliography, a list of notation, and an index for reference.
