A collection of open problems in celebration of Imre Leader's 60th birthday

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics

The time of graph bootstrap percolation

Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding to it all new edges $e$ such that $G_t \cup e$ contains a new copy of $H$. We say that $G$ percolates if for some $t \geq 0$, we have $G_t = K_n$. For $H = K_r$, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollob\'as and Morris considered graph bootstrap percolation for $G = G(n,p)$ and studied the critical probability $p_c(n,K_r)$, for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time $t$ for all $1 \leq t \leq C \log\log n$.Comment: 18 pages, 3 figure

Orthogonal polarity graphs and Sidon sets

Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erd\H{o}s from 1938. One of the most important families of $C_4$-free graphs are the Erd\H{o}s-R\'enyi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erd\H{o}s-R\'enyi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erd\H{o}s-R\'enyi orthogonal polarity graph.Comment: The authors would like to thank Jason Williford for noticing an error in the proof of Theorem 1.2 in the previous version. This error has now been correcte