arXiv subjects
Publications and source records attributed to Ayush Basu.
We extend the hypergraph regularity method to sparse $3$-uniform hypergraphs whose vertex links are $C_4$-free. In other words, we consider hypergraphs $H=(V,E)$ that are $K_{1,2,2}$-free, which implies that $|E|=O(|V|^{5/2})$. For such hypergraphs we establish a sparse analogue of the removal lemma for the tight cycle on seven vertices minus an edge.
Extending earlier results of Ne\v{s}et\v{r}il and R\"odl [Selective graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 181--189], we show that for every ordered graph $F$ there exist an ordered graph $H$ and a system $\mathscr{H}_F$ of induced copies of $F$ such that every colouring of the edges of $H$ yields a canonically coloured copy of $F$ from $\mathscr{H}_F$ and any two copies from $\mathscr{H}_F$ intersect either in a vertex or an edge or not at all. As a consequence, this allows us to construct, for any given ordered graph $F$, canonical Ramsey graphs $H$ enjoying additional structural properties. In particular, $H$ can have the same clique number as $F$ and, provided $F$ is not bipartite, the same odd girth. Moreover, if $F$ is connected, then the copies of $F$ from $\mathscr{H}_F$ are not only induced, but their pairs of vertices also have the same distances in $H$ as in $F$.
A $k$-uniform hypergraph $H$ is called a partial $(k,\ell)$-system if every set of $\ell$ vertices of $V(H)$ is contained in at most one edge of $H$. We prove the existence of a partial $(k,k-1)$-system $H$ whose Ramsey number with $r \geq 4$ colors grows as a tower of height $k-1$.
We study quantitative aspects of the following fact: For every graph $F$, there exists a graph $G$ with the property that any $2$-coloring of the triangles of $G$ yields an induced copy of $F$, in which all triangles are monochromatic. We define the Ramsey number $R_{\text{ind}}^{\Delta}(F)$ as the smallest size of such a graph $G$. Although this fact has several proofs, all of them provide tower-type bounds. We study the number $R_{\text{ind}}^{\Delta}(F)$ for some particular classes of graphs $F$.
We study the maximum number of $r$-vertex cliques in $(r-1)$-uniform hypergraphs not containing complete $r$-partite hypergraphs $K_r^{(r-1)}(a_1, \dots, a_r)$. By using the hypergraph removal lemma, we show that this maximum is $o( n^{r - 1/(a_1 \cdots a_{r-1})} )$. This immediately implies the corresponding results of Mubayi and Mukherjee and of Balogh, Jiang, and Luo for graphs. We also provide a lower bound by using hypergraph Tur\'an numbers.
The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In particular, for $k=1$ this notion is equivalent to the clique number of a graph $G$. Extending results of Frieze and Reed, and Eaton and Grable, we study the $k$-representation number of $G(n,1/2)$. As a tool, we will prove a sharp concentration result counting the number of induced subgraphs of $G(n,1/2)$ with density $(\frac{1}{2}+\alpha)$. In Lemma 3.7, we will show that the number of such subgraphs is close to its expected value with probability $1-\exp(-n^C)$.
In their classical paper, Erd\H{o}s, Goodman and P\'{o}sa studied the representation of a graph with vertex set $[n]$ by a family of subsets $S_1,\dots, S_n$ with the property that $\{i,j\}$ is an edge if and only if $S_i\cap S_j\neq \emptyset$. In this note, we consider a similar representation of bounded degree $r$-uniform hypergraphs and establish some bounds for a corresponding problem.