Tournaments, 4-uniform hypergraphs, and an exact extremal result

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a `switching' operation on tournaments that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure

A sharp threshold for a modified bootstrap percolation with recovery

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: `healthy' or `infected' and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.Comment: 45 page

Lower bounds for bootstrap percolation on Galton-Watson trees

Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.Comment: 7 page

The Health Consequences of Senior Hunger in the United States: Evidence from the 1999-2010 NHANES

Millions of seniors are food insecure in the United States, meaning that scores do not have access to enough food at all times for an active, healthy life. What makes food insecurity an even more pressing issue is its association with a wide array of negative nutrition and health consequences. In earlier reports on food insecurity among seniors it was documented that food insecure seniors, even after controlling for other factors, were at higher risk of experiencing negative nutrition and health consequences than food secure seniors. In this report, we build on those earlier findings in three main directions. Namely, we add in several new health outcomes; we use four more years of data ; and we examine how trends in health and nutrition outcomes among food secure and food insecure seniors have changed over the past decade. Using data from the 1999-2010 National Health and Nutrition Examination Survey (NHANES), we considered the following outcomes related to nutrient intakes: energy intake, protein, vitamin A, vitamin C, thiamin, riboflavin, vitamin B6, calcium, phosphorous, magnesium, and iron. The set of health outcomes we analyzed were diabetes, general health , depression, diabetes, ADL limitations, high blood pressure, high cholesterol, congestive heart failure, coronary heart disease, cancer, reports of chest pain, gum disease, psoriasis, asthma, having had a heart attack, and a self-report of gum health

Understanding How Farmers Evaluate and Value Services

Farm Management,

Extremal numbers for odd cycles

We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.Comment: 6 page

Do Education Decisions Respond to Returns by Field of Study?

We utilize the 2000 cohort of university graduates from the National Graduate Survey (NGS) to estimate the extent to which the choice of field of study is influenced by expected returns to those fields of study. The expected returns are based on earnings equations estimated from the earlier 1990 NGS cohort for the years 1992 and 1995 -- years that are around the time when the 2000 cohort would be applying to university and forming expectations of their expected returns by field of study. We estimate those expected returns using conventional OLS earnings equations as well as IV estimates to account for the potential endogeneity of the returns by field of study since selection effects may bias the expected returns. Our IV estimates utilize measures of skill-biased technological change as instruments. Overall, our results suggest that prospective students do choose fields of study in part at least on the basis of earnings they can expect to receive in those fields. Furthermore, earnings expectations formed around the time they are applying are more influential than earnings expectations based on years further away from that time, although both generally have an impact on the choice of field of study.Education decisions; field of study; returns to education; multi-nomial logits; National Graduate Survey (NGS)