Local rainbow colorings for various graphs

Motivated by a problem in theoretical computer science suggested by Wigderson, Alon and Ben-Eliezer studied the following extremal problem systematically one decade ago. Given a graph $H$, let $C(n,H)$ be the minimum number $k$ such that the following holds. There are $n$ colorings of $E(K_{n})$ with $k$ colors, each associated with one of the vertices of $K_{n}$, such that for every copy $T$ of $H$ in $K_{n}$, at least one of the colorings that are associated with $V(T)$ assigns distinct colors to all the edges of $E(T)$. In this paper, we obtain several new results in this problem including: \begin{itemize} \item For paths of short length, we show that $C(n,P_{4})=\Omega(n^{\frac{1}{5}})$ and $C(n,P_{t})=\Omega(n^{\frac{1}{3}})$ with $t\in\{5,6\}$, which significantly improve the previously known lower bounds $(\log{n})^{\Omega(1)}$. \item We make progress on the problem of Alon and Ben-Eliezer about complete graphs, more precisely, we show that $C(n,K_{r})=\Omega(n^{\frac{2}{3}})$ when $r\geqslant 8$. This provides the first instance of graph for which the lower bound goes beyond the natural barrier $\Omega(n^{\frac{1}{2}})$. Moreover, we prove that $C(n,K_{s,t})=\Omega(n^{\frac{2}{3}})$ for $t\geqslant s\geqslant 7$. \item When $H$ is a star with at least $4$ leaves, a matching of size at least $4$, or a path of length at least $7$, we give the new lower bound for $C(n,H)$. We also show that for any graph $H$ with at least $6$ edges, $C(n,H)$ is polynomial in $n$. All of these improve the corresponding results obtained by Alon and Ben-Eliezer.Comment: 19 page

Euclidean Gallai-Ramsey for various configurations

The Euclidean Gallai-Ramsey problem, which investigates the existence of monochromatic or rainbow configurations in a colored $n$-dimensional Euclidean space $\mathbb{E}^{n}$, was introduced and studied recently. We further explore this problem for various configurations including triangles, squares, lines, and the structures with specific properties, such as rectangular and spherical configurations. Several of our new results provide refinements to the results presented in a recent work by Mao, Ozeki and Wang. One intriguing phenomenon evident on the Gallai-Ramsey results proven in this paper is that the dimensions of spaces are often independent of the number of colors. Our proofs primarily adopt a geometric perspective

Sperner systems with restricted differences

Let $\mathcal{F}$ be a family of subsets of $[n]$ and $L$ be a subset of $[n]$. We say $\mathcal{F}$ is an $L$-differencing Sperner system if $|A\setminus B|\in L$ for any distinct $A,B\in\mathcal{F}$. Let $p$ be a prime and $q$ be a power of $p$. Frankl first studied $p$-modular $L$-differencing Sperner systems and showed an upper bound of the form $\sum_{i=0}^{|L|}\binom{n}{i}$. In this paper, we obtain new upper bounds on $q$-modular $L$-differencing Sperner systems using elementary $p$-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the $q$-modular setting, which results in several new upper bounds on $q$-modular $L$-avoiding $L$-intersecting systems. In particular, we improve a result of Felszeghy, Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve

A new variant of the Erd\H{o}s-Gy\'{a}rf\'{a}s problem on K5K_{5}

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of given graph $H$ in which every color appears an even number of times. When $H=K_{4}$, the question of whether $n^{o(1)}$ colors are enough, was initially emphasized by Alon. Through modifications to the coloring functions originally designed by Mubayi, and Conlon, Fox, Lee and Sudakov, the question of $K_{4}$ has already been addressed. Expanding on this line of inquiry, we further study this new variant of the generalized Ramsey problem and provide a conclusively affirmative answer to Alon's question concerning $K_{5}$.Comment: Note added: Heath and Zerbib also proved the result on $K_{5}$ independently. arXiv:2307.0131