Ramsey numbers of Berge-hypergraphs and related structures

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.Comment: Updated to include suggestions of the refere

The Ramsey numbers R(K_3, K_8 - e) and R(K_3, K_9 - e)

We give a general construction of a triangle free graph on 4p points whose complement does not contain K_p+2 - e for p \u3e= 4. This implies the the Ramsey number R(K_3, K_k - e) \u3e= 4k - 7 for k \u3e= 6. We also present a cyclic triangle free graph on 30 points whose complement does not contain K_9 - e. The first construction gives lower bounds equal to the exact values of the corresponding Ramsey number for k = 6, 7 and 8. the upper bounds are obtained by using computer algorithms. In particular, we obtain two new values of Ramsey numbers R(K_3, K_8 - e) = 25 and R(K_3, K_9 - e) = 31, the bounds 36 \u3c= R(K_3, K_10 - e) \u3c= 39, and the uniqueness of extremal graphs for Ramsey numbers R(K_3, K_6 - e) and R(K_3, K_7 - e)

Path-kipas Ramsey numbers

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest positive integer $p$ such that for every graph $G$ on $p$ vertices the following holds: either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. In this paper, we study the Ramsey numbers $R(P_n,\hat{K}_m)$, where $P_n$ is a path on $n$ vertices and $\hat{K}_m$ is the graph obtained from the join of $K_1$ and $P_m$. We determine the exact values of $R(P_n,\hat{K}_m)$ for the following values of $n$ and $m$: $1\le n \le 5$ and $m\ge 3$; $n\ge 6$ and ($m$ is odd, $3\le m\le 2n-1$) or ($m$ is even, $4\le m \le n+1$); $6\le n\le 7$ and $m=2n-2$ or $m \ge 2n$; $n\ge 8$ and $m=2n-2$ or $m=2n$ or ($q\cdot n-2q+1 \le m\le q\cdot n-q+2$ with $3\le q\le n-5$) or $m\ge (n-3)^2$; odd $n\ge 9$ and ($q\cdot n-3q+1\le m\le q\cdot n-2q$ with $3\le q\le (n-3)/2$) or ($q\cdot n-q-n+4m\le q\cdot n-2q$ with $(n-1)/2\le q\le n-4).$ Moreover, we give lower bounds and upper bounds for $R(P_n ,\hat{K}_m)$ for the other values of $m$ and $n$

Ramsey numbers for trees II

Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,...,v_{n_1},w_0,w_1,...,w_{n_2}\}$ and $E(S(n_1,n_2))=\{v_0v_1,...,v_0v_{n_1},v_0w_0, w_0w_1,...,w_0w_{n_2}\}$. In this paper we determine $r(K_{1,m-1},S(n_1,n_2))$ for $n_1\ge m-2\ge n_2$. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{"}=(V,E_2)$ and $T_n^{'"} =(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0v_1,...,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},$ $v_2v_{n-1}\}$ and $E_3=\{v_0v_1,...,v_0v_{n-4},v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$. In this paper we obtain explicit formulas for $r(K_{1,m-1},T_n)$ $(n>m+3)$, $r(T_m',T_n)$ $(n>m+4)$, $r(T_n,T_n)$, $r(T_n',T_n)$ and $r(P_n,T_n)$, where $T_n\in\{T_n",T_n'",T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n'$ is the unique tree with $n$ vertices and maximal degree $n-2$.Comment: add two conjecture

The First Classical Ramsey Number for Hypergraphs is Computed

With the help of the computer, we have shown that in any coloring with two colors of the triangles on a set of 13 points there must exist a monochromatic tetrahedron. This proves the new upper bound R (4,4;3) \u3c = 13. The previous best upper bound of 15 was derived independently by Giraud (1969 [2]), Schwenk (1978 [5]) and Sidorenko (1980 [6]). The first construction of a R (4,4;3)-good hypergraph on 12 points was presented by Isbell (1969 [3]), and the same one again more elegantly by Sidorenko (1980 [6]). We have constructed more than 200,000 R (4,4;3)-good hypergraphs on 12 points, but probably not the full set. R (4,4;3)=13 is the first known exact value of a classical Ramsey number for hypergraphs. The solution was achieved with the help of a variety of algorithms relying on a strong connection between the colorings with two colors of the triangles on n points and the so-called Tura´n set systems T(n ,5,4). The main criterion used to prune the search space for R (4,4;3)-good hypergraphs was to count the number of 4-sets containing two triangles of each color; such families of 4-sets are known to form Tura´n systems and their cardinalities must be minorized by the corresponding Tura´n numbers T(n ,5,4). We used an innovative method for generating large families of set systems which efficiently prevents isomorphic copies of set systems being produced. This method has many potential applications to other general computer searches for elusive combinatorial configurations. As a check on the correctness of the algorithms, many of the intermediate subfamilies of R (4,4;3)-good hypergraphs were generated by two different methods: from colorings of triangles on a smaller number of points and independently via Tura´n systems. An important component of the software used was a general set-system automorphism group program [4]