Ramsey numbers and the size of graphs

For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as a function of m

Small Complete Minors Above the Extremal Edge Density

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest real c(t) so that every n-vertex graph with c(t)n edges contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of order at most C(\epsilon)log n. We use our extension of Mader's theorem to prove that such a graph G must contain a K_t-minor of order at most C(\epsilon)log n loglog n. Known constructions of graphs with high girth show that this result is tight up to the loglog n factor

The minimum number of nonnegative edges in hypergraphs

An r-unform n-vertex hypergraph H is said to have the Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n>10k^3 real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture which states that for n >= 4k and any weighting on the 1-dimensional subspaces of F_q^n with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado theorem on k-intersecting families

Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length l for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph on n > 4k^4 vertices with independence number k, then G is pancyclic. He then suggested that n = \Omega(k^2) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > ck^{7/3} suffices