Bounds on Ramsey Games via Alterations

This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdos and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.Comment: 9 page

Online Ramsey Numbers and the Subgraph Query Problem

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 $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ 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 improvement $$\tilde{r}(n,n) \ge 2^{(2-\sqrt{2})n + O(1)}$$ for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement $$\tilde{r}(m,n) \ge n^{(2-\sqrt{2})m + O(1)}$$ for the off-diagonal case, where $m\ge 3$ is fixed and $n\rightarrow\infty$. Using a different randomized Painter strategy, we prove that $\tilde{r}(3,n)=\tilde{\Theta}(n^3)$, determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for $m \geq 4$. 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 Erd\H{o}s--R\'{e}nyi 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.Comment: Corrected substantial error in the proof of Theorem

Note on down-set thresholds

Gunby-He-Narayanan showed that the logarithmic gap predictions of Kahn-Kalai and Talagrand (proved by Park-Pham and Frankston-Kahn-Narayanan-Park) about thresholds of up-sets do not apply to down-sets. In particular, for the down-set of triangle-free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down-sets of F-free graphs.Comment: 5 pages; to appear in Random Structures and Algorithms (RSA

Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

Duke vs Clemson (10/20/1962)

Duke vs Clemson (10/20/1962)https://tigerprints.clemson.edu/fball_prgms/1055/thumbnail.jp

Online Ramsey Numbers and the Subgraph Query Problem

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 K_m or a blue K_n using as few turns as possible. The online Ramsey number [equation; see abstract in PDF for details] 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 improvement [equation; see abstract in PDF for details] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement [equation; see abstract in PDF for details] for the off-diagonal case, where m ≥ 3 is fixed and n → ∞. Using a different randomized Painter strategy, we prove that [equation; see abstract in PDF for details], 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

The Crescent - October 15, 1954

Volume 66, Number 2https://digitalcommons.georgefox.edu/the_crescent/1619/thumbnail.jp

Xavier University Newswire

https://www.exhibit.xavier.edu/student_newspaper/4070/thumbnail.jp

Algorithmic approaches to problems in probabilistic combinatorics

The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method: 1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse. 2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n). 3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory. 4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D