Anti-Ramsey numbers of small graphs

The anti-Ramsey number $AR(n,G$), for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy of $G$ whose edges have distinct colours. In this paper we determine the anti-Ramsey numbers of all graphs having at most four edges

On the degree of regularity of some equations

AbstractIn this paper we investigate the behaviour of the solutions of equations ΣI=1n aixi = b, where Σi=1n, ai = 0 and b ≠ 0, with respect to colorings of the set N of positive integers. It turns out that for any b ≠ 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 − x2 = x2 − x1 + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochromatic solution. For b even and 2-colorings, always monochromatic solutions can be found, and bounds for the corresponding Rado numbers are given. If one imposes the ordering x1 < x2 < x3, then there exists already a 4-coloring of N, which prevents a monochromatic solution of x3 − x2 = x2 − x1 + b, where b ϵ N

Ramsey theory, Discrepancy Theory, Zero-Sums and Symmetric Functions

The underlying philosophy of Ramsey Theory is that total disorder is impossible, and the underlying philosophy of Discrepancy Theory is that a totally equal distribution is impossible. In both theories we mainly try to find extremal configurations that satisfy some property or at least their magnitude. It is convenient to express these configurations as colored objects. Both theories can be extended from using colors to the use of vanishing (or almost vanishing) linear sums in several variables. Theorems in Ramsey Theory can be generalized using the ErdÅ s Ginzburg Ziv theorem, by replacing a {0,1}-coloring by a coloring which uses the residues modulo a positive integer assuring a modular zero-sum. In Discrepancy Theory, many combinatorial problems can be expressed by a {-1,1}-coloring and the discrepancy from a uniform distribution is expressed as the deviation from zero. In the lecture we will discuss from a personal perspective, several Ramsey-type theorems and Discrepancy theorems in order to demonstrate the breadth of the subjects. Next we will survey recent developments of the EGZ theorem and other developments relating to integer-coloring. Finally, we will show how these developments relate to Ramsey Theory and Discrepancy Theory. The linear sums mentioned above can be generalized to symmetric polynomials. This suggests new avenues of research and many more problems.Non UBCUnreviewedAuthor affiliation: University of IdahoOthe

SOME PROBLEMS IN VIEW OF RECENT DEVELOPMENTS OF THE Erdős Ginzburg Ziv Theorem

Two conjectures concerning the Erdős Ginzburg Ziv theorem were recently confirmed. Reiher and di Fiore proved independently the two dimension analogue of the EGZ theorem, as conjectured by Kemnitz, and Grynkiewicz proved the weighed generalization of the EGZ theorem as conjectured by Caro. These developments trigger some further problems. First, we will present computer experiments that at least for small numbers reveal very simple phenomena of zero sum theorems that seem to be difficult to prove. Next, we will examine the notion of generalization of Ramsey type theorems in the sense of a given zero sum theorem in view of the new developments

On the intersection of two m-sets and the Erdos-Ginzburg-Ziv theorem

We prove the following extension of the Erdos-Ginzburg-Ziv Theorem. Let m be a positive integer. For every sequence {ai}i∈I of elements from the cyclic group ℤm, where |I| = 4m - 5 (where |I| = 4m -3), there exist two subsets A, B ⊆ I such that \A ∩ B\ = 2 (such that \A ∩ B\ = 1), |A| = |B| = m, and Σi∈A ai = Σi∈b bi = 0