We consider the strong Ramsey-type game R (k) (H, ℵ 0 ), played on the edge set of the infinite complete k-uniform hypergraph K k N . Two players, called FP (the first player) and SP (the second player), take turns claiming edges of K k N with the goal of building a copy of some finite predetermined k-uniform hypergraph H. The first player to build a copy of H wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw.In this paper, we construct a 5-uniform hypergraph H such that R (5) (H, ℵ 0 ) is a draw. This is in stark contrast to the corresponding finite game R (5) (H, n), played on the edge set of K 5 n . Indeed, using a classical game-theoretic argument known as strategy stealing and a Ramsey-type argument, one can show that for every k-uniform hypergraph G, there exists an integer n 0 such that FP has a winning strategy for R (k) (G, n) for every n ≥ n 0 .

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We bound the second eigenvalue of random d-regular graphs, for a wide range of degrees d, using a novel approach based on Fourier analysis. Let G n,d be a uniform random d-regular graph on n vertices, and λ(G n,d ) is its second largest eigenvalue by absolute value. For some constant c > 0 and any degree d with log 10 n ≪ d ≤ cn, we show that λ(G n,d ) = (2 + o(1)) d(n − d)/n asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(G n,d ) for all d ≤ cn. To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d-regular random graphs -especially those of Liebenau and Wormald [14].

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We bound the second eigenvalue of random 𝑑-regular graphs, for a wide range of degrees 𝑑, using a novel approach based on Fourier analysis. Let G n,𝑑 be a uniform random 𝑑-regular graph on n vertices, and 𝜆(G n,𝑑 ) be its second largest eigenvalue by absolute value. For some constant c > 0 and any degree 𝑑 with log 10 n ≪ 𝑑 ≤ cn, we show that 𝜆(G n,𝑑 ) = (2 + o( 1))√ 𝑑(n − 𝑑)∕n asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of 𝜆(G n,𝑑 ) for all 𝑑 ≤ cn. To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on 𝑑-regular random graphs-especially those of Liebenau and Wormald.