The van der Waerden number w.rI k 1 ; k 2 ; : : : ; k r / is the least integer m such that for every partition P 1 [ P 2 [ [ P r of the set ¹1; 2; : : : ; mº, there is an index j in ¹1; 2; : : : ; rº such that P j contains an arithmetic progression of k j terms. We have computed exact values of the previously unknown van der Waerden numbers w.2I 3; 17/ and w.2I 3; 18/.

The van der Waerden number w.rI k 1 ; k 2 ; : : : ; k r / is the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; rº such that P j contains an arithmetic progression of length k j . We have computed exact values of some (30) previously unknown van der Waerden numbers and also computed lower bounds of others. Let w d .rI k 1 ; k 2 ; : : : ; k r / be the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; r 1º such that P j contains an arithmetic progression of length k j , or P r contains an arithmetic progression of length k r with common difference at most d . A table of observed values of w d .rI k 1 ; k 2 ; : : : ; k r / for d D 1; 2; : : : ; b m 1 kr 1 c is given.

Electrocatalytic direct seawater splitting is considered to be one of the most desirable and necessitous approach to produce substantial amount of green hydrogen to meet the energy demand. However, practically...

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

The urea oxidation reaction (UOR) with low required oxidation potential not only identified as the energy saving strategy for efficient hydrogen production but also offers an effective way to treat...

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In this paper, we are concerned with calculating $r(k, n)$, the length of the longest $k$-AP free subsequences in $1, 2, \ldots , n$. We prove the basic inequality $r(k, n) \le n − \lfloor m/2\rfloor$, where $n = m(k − 1) + r$ and $r < k − 1$. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdős and Turán) and describe a simple greedy algorithm that appears to give an optimal $k$-AP free sequence infinitely often. We provide many exact values of $r(k, n)$ in the Appendix.

Production of green hydrogen from the electrolysis of water is considered to be one of the most desirable processes to address the clean energy demand. However, a high energy barrier and an economically unsustainable nature limit the process toward practical implementation in an extensive way. The designing of an electrocatalyst accompanied by defect engineering, heteroatom doping, and strain creation are known to be effective strategies to make the process efficient. Recently, construction of heterojunctions with a combination of materials with desired band structures is considered to be a smart approach in promoting electrocatalytic activities by attaining charge redistribution and manipulating the electronic structure at the interface. The present review elaborately discusses possible heterojunctions, alterations at the interface, insight thermodynamics, and possible causes for boosting electrocatalytic activities. The heterojunction-based electrocatalysts are found not only effective for oxygen evolution reactions and hydrogen evolution reactions, but also to be very useful toward various electrochemical oxidation reactions as well as electrochemical reduction reactions involving small molecules undergoing decomposition at very low energy. Having such a specialty, there is no doubt that heterojunction-based electrocatalysts are going to be the primary choices of researchers in coming years.