Abstract: In the talk I will present two short results:
(a) Define $T=T(k)$ the minimal $t$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of the numbers $1,\dots, tn$ for all $n$, where equinumerous means that each color used the same number of times. Almost answering a question of Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic, we almost determine the function $T$. It is a joint work with Linz.
(b) Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobas in 1968. In this process, we start with initial "infected" set of edges $E(0)$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E(t)$ subset of $E(K_n)$, we infected a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E(t) + e)$. A question raised by Bollobas asks for the maximum time the process can run before it stabilizes. In 2015, Bollobas, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H$ is the $r$-vertex complete graph. They answered the question for $r > 3$ and gave a lower bound for every $r \ge 5$. In their paper, they also conjectured that the maximal running time is subquadratic for every integer $r$. In this paper we disprove their conjecture for every $r$ at least 6 and we give a better lower bound for the case that $r=5$. In the proof of the case $r=5$ we use the Behrend construction. Joint result with Kronenberg, Pokrovskiy and Szabo.