The authors would like to thank Lap Chi Lau for useful discussions, and in particular his suggestion to look at [LSW15]. Talya Eden is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.

For any X-nef and big divisor class D on a surface X of type P27,0, one has

Our final goal is to sketch briefly a justification for Conjecture 1.1 without reference to the function field results in the body of the paper. In addition, we indicate how to generate a conjecture for the lower order terms in the asymptotic expans…

In this section we explore the design space of hollow heaps. We show that our data structures occupy “sweet spots” in the design space: although small changes to these structures preserve their efficiency, larger changes destroy it. We explore varia…

We have investigated several interesting graph properties (minimum degree, vertex-connectivity, Hamiltonicity) of the trace of a long-enough random walk on a dense-enough random graph, showing that in the relevant regimes, the trace behaves much lik…

A natural question that extends the above is to try and determine Rreg(k,ℓ) for other values of k and ℓ (in particular for fixed values of k>3 and ℓ arbitrary large), and also to try and investigate its relation with R(k,ℓ). The following conjecture…

In this section we prove our main result. Recall that the expected matching size of π (with respect to μ) is Eμ[|^M(Π)∩E(G)|]. We shall prove the following theorem.

We thank Ely Kovetz and Rennan Barkana for their useful comments on an earlier version of the draft. We also thank an anonymous referee for useful comments. This work was supported by the TAU-UCSD Cosmology Program, and by grants from the US-Israel …

(1) We note that most of the proof of Theorem 2 can be carried out over the complex domain. The only place where we (implicitly) use the fact that the underlying field is R is in the final step, where we apply the bound of Guth and Katz, as a “black…

Let E2n and O2n denote the sets of even and odd elements of Z2n, respectively, and recall that O2n is the unique maximum-size sum-free subset of Z2n. Let G=Z2n and recall that Gp denotes the p-random subset of G. We shall prove Theorem 1.1, which gi…