The (m, n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red Km or a blue Kn using as few turns as possible. The online Ramsey numberr(m, n) is the minimum number of edges Builder needs to guarantee a win in the (m, n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvementfor the lower bound on the diagonal online Ramsey number, as well as a corresponding improvementfor the off-diagonal case, where m ≥ 3 is fixed and n → ∞. Using a different randomized Painter strategy, we prove thatr(3, n) =Θ(n 3 ), determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for m ≥ 4.In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown Erdős-Rényi random graph G(N, p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem. IntroductionThe Ramsey number r(m, n) is the minimum integer N such that every red/blue-coloring of the edges of the complete graph K N on N vertices contains either a red K m or a blue K n . Ramsey's theorem guarantees the existence of r(m, n) and determining or estimating Ramsey numbers is a central problem in combinatorics. Classical results of Erdős-Szekeres and Erdős imply that 2 n/2 ≤ r(n, n) ≤ 2 2n for n ≥ 2. The only improvements to these bounds over the last seventy years have been to lower order terms (see [9,26]), with the best known lower bound coming from an application of the Lovász local lemma [14].Off-diagonal Ramsey numbers, where m is fixed and n tends to infinity, have also received considerable attention. In progress that has closely mirrored and often instigated advances on the probabilistic method, we now know that r(3, n) = Θ(n 2 / log n). The lower bound here is due to Kim [21] and the upper bound to Ajtai, Komlós and Szemerédi [1]. Recently, Bohman and Keevash [8] and, independently, Fiz Pontiveros, Griffiths and Morris [18] improved the constant √ 2 )n in Theorem 1, we get the following immediate corollary. Corollary 2. For the diagonal online Ramsey numbersr(n),As for the off-diagonal case, when m is fixed and n → ∞, Theorem 1 can be also used to substantially improve the best-known lower bound. In this case, we take c ≈ (1 − 1 √ 2 )m, d = 0, and p = C m log n n for a sufficiently large C > 0 to obtain the following corollary.

Recently, Farnik asked whether the hat guessing number $\mathrm{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\mathrm{HG}(G)\ge 2^d$ is possible. We show that for all $d\ge 1$ there exists a $d$-degenerate graph $G$ for which $\mathrm{HG}(G) \ge 2^{2^{d-1}}$. We also give a new general method for obtaining upper bounds on $\mathrm{HG}(G)$. The question of whether $\mathrm{HG}(G)$ is bounded as a function of $d$ remains open.

The hat-guessing number is a graph invariant defined by Butler, Hajiaghayi, Kleinberg, and Leighton. We determine the hat-guessing number exactly for book graphs with sufficiently many pages, improving previously known lower bounds of He and Li and exactly matching an upper bound of Gadouleau. We prove that the hat-guessing number of K 3,3 is 3, making this the first complete bipartite graph K n,n for which the hat-guessing number is known to be smaller than the upper bound of n `1 of Gadouleau and Georgiou. Finally, we determine the hat-guessing number of windmill graphs for most choices of parameters.

Given a fixed graph H, a real number p ∈ (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in n δ queries.

Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.