Cycles are strongly Ramsey-unsaturated

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method

Asymmetric Ramsey properties of random graphs involving cliques and cycles

We prove that for every ℓ, r ≥ 3, there exists c > 0 such that for (image found), with high probability there is a 2-edge-colouring of the random graph Gn,p with no monochromatic copy of Kr of the first colour and no monochromatic copy of Cℓ of the second colour. This is a progress on a conjecture of Kohayakawa and Kreuter

Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles

Confirming a conjecture of Gy´arf´as, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmikl´ossy [Discrete Math., 340 (2017), pp. 2053–2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number

Maximum number of triangle-free edge colourings with five and six colours

Let k ≥ 3 and r ≥ 2 be natural numbers. For a graph G, let F(G, k, r) denote the number of colourings of the edges of G with colours 1,…, r such that, for every colour c ∈ {1,…, r}, the edges of colour c contain no complete graph on k vertices Kk. Let F(n, k, r) denote the maximum of F(G, k, r) over all graphs G on n vertices. The problem of determining F(n, k, r) was first proposed by Erdős and Rothschild in 1974, and has so far been solved only for r = 2; 3, and a small number of other cases. In this paper we consider the question for the cases k = 3 and r = 5 or r = 6. We almost exactly determine the value F(n, 3, 6) and approximately determine the value F(n, 3, 5) for large values of n. We also characterise all extremal graphs for r = 6 and prove a stability result for r = 5

Concentration Dependence of Superconductivity and Order-Disorder Transition in the Hexagonal Rubidium Tungsten Bronze RbxWO3. Interfacial and bulk properties

We revisited the problem of the stability of the superconducting state in RbxWO3 and identified the main causes of the contradictory data previously published. We have shown that the ordering of the Rb vacancies in the nonstoichiometric compounds have a major detrimental effect on the superconducting temperature Tc.The order-disorder transition is first order only near x = 0.25, where it cannot be quenched effectively and Tc is reduced below 1K. We found that the high Tc's which were sometimes deduced from resistivity measurements, and attributed to compounds with .25 < x < .30, are to be ascribed to interfacial superconductivity which generates spectacular non-linear effects. We also clarified the effect of acid etching and set more precisely the low-rubidium-content boundary of the hexagonal phase.This work makes clear that Tc would increase continuously (from 2 K to 5.5 K) as we approach this boundary (x = 0.20), if no ordering would take place - as its is approximately the case in CsxWO3. This behaviour is reminiscent of the tetragonal tungsten bronze NaxWO3 and asks the same question : what mechanism is responsible for this large increase of Tc despite the considerable associated reduction of the electron density of state ? By reviewing the other available data on these bronzes we conclude that the theoretical models which are able to answer this question are probably those where the instability of the lattice plays a major role and, particularly, the model which call upon local structural excitations (LSE), associated with the missing alkali atoms.Comment: To be published in Physical Review

Primary resistance to cetuximab therapy in EGFR FISH-positive colorectal cancer patients

The impact of KRAS mutations on cetuximab sensitivity in epidermal growth factor receptor fluorescence in situ hybridisation-positive (EGFR FISH+) metastatic colorectal cancer patients (mCRC) has not been previously investigated. In the present study, we analysed KRAS, BRAF, PI3KCA, MET, and IGF1R in 85 mCRC treated with cetuximab-based therapy in whom EGFR status was known. KRAS mutations (52.5%) negatively affected response only in EGFR FISH+ patients. EGFR FISH+/KRAS mutated had a significantly lower response rate (P=0.04) than EGFR FISH+/KRAS wild type patients. Four EGFR FISH+ patients with KRAS mutations responded to cetuximab therapy. BRAF was mutated in 5.0% of patients and none responded to the therapy. PI3KCA mutations (17.7%) were not associated to cetuximab sensitivity. Patients overexpressing IGF1R (74.3%) had significantly longer survival than patients with low IGF1R expression (P=0.006), with no difference in response rate. IGF1R gene amplification was not detected, and only two (2.6%) patients, both responders, had MET gene amplification. In conclusion, KRAS mutations are associated with cetuximab failure in EGFR FISH+ mCRC, even if it does not preclude response. The rarity of MET and IGF1R gene amplification suggests a marginal role in primary resistance. The potential prognostic implication of IGF1R expression merits further evaluation

Monochromatic cycles in 2-coloured graphs

Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length ℓ, for all ℓ ∈ [4, ⌈n/2⌉]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G)=3n/4 that do not contain all such cycles. Finally, we show that, for all δ>0 and n>n 0(δ), if G is a 2-edge coloured graph of order n with δ(G) ≥ 3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+δ/2)n, or contains monochromatic cycles of all lengths ℓ ∈ [3, (2/3-δ)n]