Proof of Lemma 3. 1. 1. For star networks, the chance of game continuation is always decreasing in n, but the expected repayment can be increasing in n if ↵ is not too large. Q Q s n+1 n , ↵ + Xi=1 ↵n+1—i (1 — ↵)i ✓n◆ .ai Σn—i + Øi—1 Σn+1—iΣ X n .
Proof of Lemma 3. 3.2 For this proof, we will define tailored linear orders for every subset of V = V (G) and use them to bound the number of independent sets. Roughly speaking we first show that under such linear orders, any independent set I, |I| ≥ q + 1, admits a unique q-prefix, which is a sequence of q distinct elements from I we define below. We later establish an upper bound on the number of independent sets of size q + m that have the same q-prefix. It will be convenient to assume (without loss of generality) that V ⊂ [n]. For any V r ⊂ V , let max(V r) be the vertex of maximum degree in G[V r] of largest value as a number in [n]. With this definition we may construct a linear order >V ′ of V r as v1, v2, . . . , vm, m = |V r|, by setting v1 = max(V r) and vi+1 = max V r \ {v1, . . . , vi} for 1 ≤ i < m. Given any independent set I in G with |I| ≥ q + 1, the q-prefix of I is a sequence (v1, . . . , vq) constructed as follows. Let v1 denote the first element of I in the linear order of >V1 , where V1 = V . For 1 ≤ i ≤ q let Vi+1 = {v ∈ Vi : : : vi >Vi v, v ∈/ NG(vi)} (3.29) and let vi+1 denote the first element of I \ {v1, . . . , vi} that appears in Vi+1 in the order >Vi+1 . To check that the procedure above does not stop before producing the desired sequence of length q, we prove by induction that I \ {v1, . . . , vi} ⊂ Vi+1, for i = 1, . . . , q. (3.30) Indeed, v1 is chosen in such a way that v1 >V1 w for all v1 =/ w ∈ I, thus I \ {v1} ⊂ V2. If (3.30) holds for i = j, 1 ≤ j < q, then for every w ∈ I \ {v1, . . . , vj} ⊂ Vj+1, w vj+1, we have vj+1 >Vj+1 w and w ∈/ NG(vj+1). Hence w ∈ Vj+2 and (3.30) holds for i = j + 1. m
Proof of Lemma 3. 3.4 Let A ⊂ [n] \ S, |A| ≥ 16n/s2, be an arbitrary subset. Consider the auxiliary labeled bipartite graph Γ defined as follows. The vertex classes of Γ are A and a disjoint copy of [3n]. The edge set of Γ is defined as E(Γ) = (x, u) ∈ A × [3n]: : : u = x + a1 + a2 for some a1, a2 ∈ S, a1 =/ a2}. Note that because S is a B3-set, for fixed x, u, there is at most one solution A ˜ −S 2 2 {a1, a2} for u = x + a1 + a2 with a1, a2 ∈ S. We will now argue that the multiplicity of a pair {x, y} ∈ in the multigraph CGS, which by definition is R (y x), is given by the number of two-paths connecting x to y in Γ. Indeed, there is a bijection between pairs {a , a }, {b , b } ∈ S 2 satisfying y − x = a1 + a2 − b1 − b2 and paths xuy in Γ, where u = x + a1 + a2 = y + b1 + b2. Recalling the definition of RS(·) in (3.15), it follows that RS(y − x) equals the number of two-paths connecting x to y in Γ. Consequently, e C˜GS
Proof of Lemma 3. 6. It suffices to use Lemma 3.8 and induct on decreasing m.
Proof of Lemma 3. 1.1: First, we take a combination of twists for which the nonholo- morphic part of H(z) is annihilated. Let p be in S−. We have f (z 1 ) := ( 2 H(z) − .−1 Σ H + 16√π Γ(− 2 , 4πn y)q . (p· ) (z)) p ∞ = Σ 1 . n=1 1 − H(n)q .−n ΣΣ p n 1 Σ 1 p|n 2 −n2 2 p p Note that the coefficient of qn in the holomorphic part of f (z)is 1 H(n) if p|n, H(n) if (−n) = −1, and 0 if .−n Σ = 1. The nonholomorphic part of f is supported p on multiples of p, because twisting the nonholomorphic part by the Xxxxxxxx symbol annihilates those coefficients. To eliminate what remains of the nonholomorphic part and the multiples of p in the holomorphic part, we take the twist f( · )2 . Repeating the above steps for every p ∈ S+ ∪ S−, we obtain a form which is supported on n for which the primes in S+ ∪ S− split or are inert in Q(√−n) as desired. q To obtain a modular form which is supported on coefficients which are multiples of the primes in S0, let d be the product of the primes in S0. We apply the U (d), operator, then twist by .−n Σ2 for each q ∈ S , and then apply the V (d) operator.
