Most Probably Intersecting Families of Subsets

Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1

Constructing Union-Free pairs of K-Element subsets

It is proved that one can choose [1/2(n/k)] disjoint pairs of k-element subsets of an n-element set in such a way that the unions of the pairs are all different, supposing that n > n(k)

Consumer Durable Spending

macroeconomics, consumer, durable spending

The role of search engine optimization in search marketing

This paper examines the impact of search engine optimization (SEO) on the competition between advertisers for organic and sponsored search results. The results show that a positive level of search engine optimization may improve the search engine's ranking quality and thus the satisfaction of its visitors. In the absence of sponsored links, the organic ranking is improved by SEO if and only if the quality provided by a website is sufficiently positively correlated with its valuation for consumers. In the presence of sponsored links, the results are accentuated and hold regardless of the correlation. When sponsored links serve as a second chance to acquire clicks from the search engine, low-quality websites have a reduced incentive to invest in SEO, giving an advantage to their high-quality counterparts. As a result of the high expected quality on the organic side, consumers begin their search with an organic click. Although SEO can improve consumer welfare and the payoff of high-quality sites, we find that the search engine's revenues are typically lower when advertisers spend more on SEO and thus less on sponsored links. Modeling the impact of the minimum bid set by the search engine reveals an inverse U-shaped relationship between the minimum bid and search engine profits, suggesting an optimal minimum bid that is decreasing in the level of SEO activity. © 2013 INFORMS

Color the cycles

The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013