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1. We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph onnvertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at leasts. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants 0 < A < Bsuch that (under optimal play) Proposer wins with high probability if, while Decider wins with high probability if. This is a factor oflarger than the lower bound coming from the off‐diagonal Ramsey numberr(3,s).

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2. Abstract

Letfs, k(n)be the maximum possible number ofs‐term arithmetic progressions in a set ofnintegers which contains nok‐term arithmetic progression. For all fixed integersk > s ≥ 3, we prove thatfs, k(n) = n2 − o(1), which answers an old question of Erdős. In fact, we prove upper and lower bounds forfs, k(n)which show that its growth is closely related to the bounds in Szemerédi's theorem.

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3. An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs areequivalentif there is an isomorphism between them preserving the edge‐ordering. Theedge‐ordered Ramsey number redge(H; q) of an edge‐ordered graphHis the smallestNsuch that there exists an edge‐ordered graphGonNvertices such that, for everyq‐coloring of the edges ofG, there is a monochromatic subgraph ofGequivalent toH. Recently, Balko and Vizer proved thatredge(H; q) exists, but their proof gave enormous upper bounds on these numbers. We give a new proof with a much better bound, showing there exists a constantcsuch thatfor every edge‐ordered graphHonnvertices. We also prove a polynomial bound for the edge‐ordered Ramsey number of graphs of bounded degeneracy. Finally, we prove a strengthening for graphs where every edge has a label and the labels do not necessarily have an ordering.

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4. Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$ . We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$ . We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$ . For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$ , where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$ . This solves a problem of Hebbinghaus and Srivastav.
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8. Abstract May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4– o (1) but not O ( δ 4 ). Let M ( δ ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ | G | 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” ( x , y ), ( x + d , y ), ( x , y + d ) ∈ A is at least ( M ( δ )–∊)| G | 2 . As a corollary via a recent result of Mandache, we conclude that M ( δ ) = δ 4– o (1) and M ( δ ) = ω ( δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N , there exists A ⊆ [ N ] 3 with | A | ≥ δN 3 such that for every d ≠ 0, the number of corners ( x , y , z ), ( x + d , y , z ), ( x , y + d , z ), ( x , y , z + d ) ∈ A is at most δ c log(1/ δ ) N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.
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