Upper tails for triangles

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) \Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}]. This is tight up to the value of $C_{\eta}$.Comment: 10 page

Subsampling in Smoothed Range Spaces

We consider smoothed versions of geometric range spaces, so an element of the ground set (e.g. a point) can be contained in a range with a non-binary value in $[0,1]$. Similar notions have been considered for kernels; we extend them to more general types of ranges. We then consider approximations of these range spaces through $\varepsilon$-nets and $\varepsilon$-samples (aka $\varepsilon$-approximations). We characterize when size bounds for $\varepsilon$-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for $\varepsilon$-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16 pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer International Publishing, 201

Hitting time results for Maker-Breaker games

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property \mP. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches minimum degree $2k$; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree $2$; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree $4$. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail

The Law of Social Entrepreneurship – Creating Shared Value through the Lens of Sandra Day O’Connor’s iCivics

This article calls for harmonizing state law legislation on social enterprises, due to the potential discrepancy between the various states on the nature and legal structure of social enterprises. Since 2008, legislators in thirty-five (35) states across the United States of America and the District of Columbia, have enacted some form of innovative social enterprise legislation. This new revolution in corporate law is called social entrepreneurship, mirroring social movements in the aftermath of the 2008 financial crisis. Public opinion has led to a shift in prevalent corporate governance theory, from current share-holder centric corporate governance to collaborative corporate governance. A new generation of entrepreneurs, corporations, shareholders and other stakeholders now work together to resist short-termism, achieve long-term value, and incorporate in their charters a deep obligation to act for the benefit of society at large. These developments are new, and to date there is no established body of precedent that judges, entrepreneurs, managers, board of directors or legal counsel can rely upon to make day-to-day decisions, as well as interpret and elucidate the governing laws. Accordingly, there will likely be different statutory interpretations of the nature and legal structure of a social enterprise by the different jurisdiction\u27s courts. Courts will be faced with determining what constitutes a social enterprise, when there is no agreed upon definition of what it means. Furthermore, a review of current literature on social entrepreneurship reveals ill-defined, fragmented, and incoherent theoretical terms of social enterprise and social entrepreneur. Harmonization of social entrepreneurship law is extremely important because of the prevalence of commercial and other public benefit transactions that extend beyond state borders. This article attempts to bridge the research gap, and propose a simple, inclusive, coherent and unified test that the courts, regardless of the jurisdiction, can use to determine what constitutes a social enterprise, while allowing flexibility to the various jurisdictions to tailor the language in the test to meet their unique needs and preferences. One of the elements of the “social enterprise” test centers on the identity of the founder, i.e., the social entrepreneur. Since it is not clear who is a social entrepreneur, this article will also propose a test to determine whether the founder of the entity is indeed a social entrepreneur or merely a social activist. Justice Sandra Day O’Connor’s iCivics example is used to illustrate the elements in the test for branding a “social entrepreneur.” It presents the Justice in a new light as an iconic American social entrepreneur and a pioneer of the civics education digital game-based learning

Dual Fiduciaries: Unicorns, Corporate Law and the New Frontier

Legal and regulatory structures influence the shift in equities in the United States from public markets to private markets, entrepreneurial opportunities and new firm formation. There is a rise in the number of “unicorn” firms, which are privately held venture-capital backed startups that are valued at $1 billion or more. The number of unicorns in the United States and overseas has grown exponentially over the last few years. This chapter discusses the rise of the unicorns and with it the increasing importance of corporate governance and fiduciary duties. There are new vertical and horizontal conflicts among common and preferred shareholders that corporate law scholars, practitioners and judges will have to grapple with

Incorporating Unicorns: An Empirical Analysis

There is a growing concern among regulators and academics about how to regulate unicorns - entities large enough to have a public impact yet remaining in the private domain. An examination of corporate charters within a selected sample of unicorn firms reveals an important finding: 97% of these entities are incorporated in Delaware. This concentration provides Delaware with significant leverage to shape regulatory frameworks, especially concerning the protection of parties who may lack the ability to safeguard their interests through contractual means. This groundbreaking discovery on the dominance of Delaware showcases a substantial deviation from incorporation trends in other business segments. While 79% of public firms and 67% of early-stage venture-backed private firms are incorporated in Delaware, only 2% of small private enterprises do so. The overwhelming preference for Delaware among unicorn firms is a distinct and unprecedented trend, raising intriguing questions about the specific factors driving this exceptional pattern. As unicorns evolve and continue to develop as market movers, Delaware\u27s position as their incorporation venue of choice will only grow in importance and relevance, especially due to the recent debate over changes to Delaware law and the competition among states to attract businesses

Delaware Beware

This article conducts an in-depth exploration of the dynamic competition among states to attract businesses and determine the legal framework governing corporations. It adopts an innovative market-centric viewpoint, treating corporate law as a product within the broader context of charter competition among U.S. states. While the scholarly spotlight has predominantly shone on publicly traded giants, this article daringly delves into uncharted territory, unraveling the intricate incorporation and governance decisions of privately held “unicorns”—those elusive venture capital-backed behemoths that silently shape the economic landscape. By unraveling the decision-making processes of where these economic powerhouses incorporate, the article challenges prevailing assumptions on horizontal and vertical competitive federalism, introducing the concept of “long-term private giant” companies. This distinctive perspective provides insights into the relocation options and incorporation choices of both large private and public firms, illuminating how these entities navigate and influence the intricate landscape of organizational structure and governance choices within the corporate domain

Alternative Venture Capital: The New Unicorn Investors

The U.S. Securities and Exchange Commission has promulgated new rules designed to harmonize and improve the patchwork exempt offering framework, protect investors and facilitate capital formation. Additionally, the U.S. Department of Labor announced that the fiduciary responsibility provisions of the Employee Retirement Income Security Act of 1974 do not prohibit fiduciaries of 401(k) and other individual account plans from investing in, and undertaking exposure to, private equity investments. These policies address the concern that retail investors are missing out on investment opportunities, due to fewer listed firms and initial public offerings, the greater role of the private market in raising money, and the rise in the number of unicorn firms. This Article details these concerns, assesses the policy changes within the broader context of private capital formation, and argues that some of them not only fail to provide a remedy, but may also induce greater harm and should not have been undertaken. Most importantly, policymakers must consider the rise of alternative venture capital ( AVC ) investors, and the ways in which those investors affect a unicorn firm, its capital needs, and the lack of disclosure of information, all of which affect future investors. Finally, this Article argues that to adequately protect retail investors, AVC investors ought to be considered when formulating policy decisions relating to investor protection and the capital formation needs of private companies. Increasingly seems like we are entering a new reality in Unicorn land. If you have raised more than $250mm & are NOT public, the presumption is you are losing WAY too much money, and you probably have sh[***]y unit economics. There was a theory that the public and private markets for tech company shares had become disconnected, that venture capitalists and (particularly) non-traditional venture investors like mutual funds and (most particularly) SoftBank were now willing to pay higher prices than the public markets were, and that when those private investors eventually tried to sell to the public markets they\u27d run into trouble