On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k) =n^{\Theta(\binom{2k}{k})}$ holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.Comment: 11 page

Triangle-Intersecting Families of Graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page

Intersecting Inequalities: Research to Reduce Inequality for Immigrant-Origin Children and Youth

This is one of a series of five papers outlining the particular domains and dimensions of inequality where new research may yield a better understanding of responses to this growing issue.Immigration has grown across all post-industrial nations, and inequality has risen at a steep rate on a variety of indicators, including income distribution, child poverty, residential segregation, and numerous academic outcomes.In this report, we see that among the children of immigrants, inequality is manifested against a backdrop of wide disparity in post-migration conditions faced by new immigrants. Indeed, immigrant groups represent some of the most and least advantaged groups in the U.S. in terms of skills, education, and assets. Many immigrant-origin students struggle academically, leaving school without acquiring the tools necessary to function effectively in the highly competitive, knowledge-intensive U.S. economy, in which limited education impedes wages and social mobility