Quasi-configurations: Building blocks for point-line configurations

We study point-line incidence structures and their properties in the projective plane. Our motivation is the problem of the existence of (n₄) configurations, still open for few remaining values of n. Our approach is based on quasi-configurations: point-line incidence structures where each point is incident to at least 3 lines and each line is incident to at least 3 points. We investigate the existence problem for these quasi-configurations, with a particular attention to 3j₄-configurations where each element is 3|4-valent. We use these quasi-configurations to construct the first (37₄) and (₄3₄) configurations. The existence problem of finding (22₄), (23₄), and (26₄) configurations remains open.